Abstract
We study a parabolic equation for the fractional p-Laplacian of order s, for \(p\ge 2\) and \(0<s<1\). We provide space-time Hölder estimates for weak solutions, with explicit exponents. The proofs are based on iterated discrete differentiation of the equation in the spirit of Moser’s technique.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
1.1 The problem
In this paper, we study the regularity of weak solutions to the nonlinear and nonlocal parabolic equation
where \(2\le p<\infty \), \(0<s<1\) and \((-\Delta _p)^s\) is the fractional p-Laplacian of order s, i.e. the operator formally defined by
Here \(\mathrm {P.V.}\) denotes the principal value in Cauchy sense. The operator \((-\Delta _p)^s\) arises as the first variation of the Sobolev-Slobodeckiĭ seminorm (see Sect. 2.1)
This operator can be seen as a nonlocal (or fractional) version of the \(p-\)Laplace operator,
since, as s goes to 1, solutions of \((-\Delta _p)^s u =0\) converge to solutions of \(-\Delta _p u =0\), once suitably rescaled. See for instance [3, Section 1.4] and [20].
Remark 1.1
(Homogeneity and scalings) It is important to notice that Eq. (1.1) is not homogeneous, i.e. if u is a solution, then \(\lambda \,u\) does not solve the same equation. Rather, it solves
On the other hand, solutions are invariant with respect to the natural scaling \((x,t)\mapsto (\lambda \,x,\lambda ^{s\,p}\,t)\), for any \(\lambda >0\). In other words, if u is a solution of (1.1), then the rescaled function
is still a solution. By combining the last two facts, we also get that
still solves (1.1). We will make a repeated use of this simple fact.
In this paper, we are concerned with the Hölder regularity for weak solutions of (1.1). More precisely, we prove that local weak solutions (see Definition 3.1 below) are locally \(\delta -\)Hölder continuous in space and \(\gamma -\)Hölder continuous in time, whenever
To the best of our knowledge, our result is the first pointwise continuity estimate for solutions of this equation.
1.2 Background and recent developments
In recent years there has been a surge of interest around the operator (1.2), after its introduction in [20]. In particular, equation (1.1) has been studied in [1, 25, 26, 31, 33, 34] and [35]. References [25, 26, 33] and [34] dealt with existence and uniqueness of solutions, together with their long time asymptotic behaviour. Similar properties for (1.1) with a general right-hand side in place of 0 are studied in [1]. In [35], some regularity of the semigroup operator generated by \((-\Delta _p)^s\) was studied. In [31], the local boundedness of weak solutions of (1.1) is proved.
Recently, in [17], a weaker pointwise regularity result was obtained for viscosity solutions of the doubly nonlinear equation
by using completely different methods. This equation and its large time behavior is related to the eigenvalue problem for the fractional p-Laplacian. A crucial difference between this equation and (1.1), is that the former is homogeneous, a feature which is not shared by our equation, as already observed in Remark 1.1. Moreover, the nonlinearity in the time derivative in (1.3) makes the notion of weak solutions less useful. It is not clear whether the methods in [17] can be adapted to the present situation or not.
In the linear or non-degenerate case, corresponding to \(p=2\), the literature on regularity is vast. We mention only a fraction of it, namely [7,8,9, 29, 30] and [32]. However, we point out that none of these results apply to our setting.
The stationary version of (1.1), i.e.,
has attracted a lot of attention, as well. The regularity of solutions has been studied for instance in [3, 4, 6, 14, 15, 18, 19, 21,22,23,24, 27] and [35]. In particular, the regularity result proved in the present paper can be seen as the parabolic version of that obtained by the first two authors and Schikorra in [4] for the stationary equation.
The local counterpart of (1.1) is the parabolic equation for the p-Laplacian
This has been intensively studied and only in the last decades has its theory reached a rather complete state. We refer to [12] and [13] for a complete account on the regularity results for this equation and some of its generalizations. At present, the best local regularity known is spatial \(C^{1,\alpha }-\)regularity for some \(\alpha >0\) (see [12, Chapter IX]) and \(C^{0,1/2}-\)regularity in time (see [2, Theorem 2.3]). None of these exponents is known to be sharp. However, due to the explicit solution
it is clear that solutions cannot be better than \(C^{1,1/(p-1)}\) in space.
1.3 Main result
The main result of our paper is the following Hölder regularity for local weak solutions of (1.1). Here, we use the following notation for parabolic cylinders
with \(B_r(x_0)\) denoting the \(N-\)dimensional ball of radius r centered at the point \(x_0\). For the precise definition of local weak solution, as well as of the spaces \(C^\delta _{x,\mathrm loc}(\Omega \times I)\) and \(C^\gamma _{t,\mathrm loc}(\Omega \times I)\), we refer the reader to Sects. 3.1 and 2.3, respectively.
Theorem 1.2
Let \(\Omega \subset {\mathbb {R}}^N\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that
Define the exponents
Then
More precisely, for every \(0<\delta <\Theta (s,p)\), \(0<\gamma <\Gamma (s,p)\), \(R>0\), \(x_0\in \Omega \) and \(T_0\) such that
there exists a constant \(C=C(N,s,p,\delta , \gamma )>0\) such that
for any \((x_1,\tau _1),\,(x_2,\tau _2)\in Q_{R/4,R^{s\,p}/4}(x_0,T_0)\).
Remark 1.3
(Comment on the time regularity) The regularity in time is almost sharp for \(s\,p\le (p-1)\). Indeed, our result in this case gives Hölder continuity for any exponent less than 1. The following example from [9] shows that solutions are not \(C^1\) in time in general. Let
where \(C\ne 0\) is chosen so that v is a local weak subsolution (see Definition 3.1) in \(B_1\times (-1,0]\). Then, if u is the unique solution (given by Theorem A.3) of
by Proposition A.6 we get \(u\ge v\) in \(B_1\times (-1,0]\). Moreover, by Proposition A.4, \(u=0\) in \(B_1\times (-1,-1/2)\). Therefore,
for \(h>0\) and \(x\in B_1\). Hence, u cannot have a continuous time derivative.
Remark 1.4
(Comments on the assumption) We have chosen to assume the global boundedness (1.4) of our weak solutions, in order to simplify the presentation. Actually, the estimate (1.6) could be proved under the weaker assumption
and
where the tail space \(L^{p-1}_{s\,p}({\mathbb {R}}^N)\) is defined by
We point out that by [31, Lemma 2.6], condition (1.8) is a natural one in order to guarantee the local boundedness (1.7). However, it is not known apriori if the quantity (1.8) is finite whenever u is a weak solution. Indeed, even if u solves the initial boundary value problem
with the boundary data g satisfying
it is not evident that this is sufficient to entail (1.8). For this reason, and to not overburden an already technical proof, we have chosen to assume the simpler condition (1.4). For completeness, in Appendix A we give some sufficient conditions assuring that our weak solutions verify (1.4), see Corollary A.5 below.
1.4 Main ideas of the paper
The idea we use to prove Theorem 1.2 is very similar to the method employed in [4] for the elliptic case: we differentiate equation (1.1) in a discrete sense and then test the differentiated equation against functions of the form
For suitable choices of \(\vartheta >0\) and \(\beta \ge 1\), this gives an integrability gain (see Proposition 4.1) of the form
for \(-1/2\le T\le 0\) and an arbitrary \(\mu >0\). By first fixing \(T=0\) and ignoring the second term in the left-hand side of (1.9), this can be iterated finitely many times in order to obtain
We can then use the second term in the left-hand side of (1.9), so to get
Thus, by using a Morrey-type embedding result, we can conclude that \(u\in C_\text {loc}^\delta \) spatially for any \(0<\delta <s\).
After this, we prove Proposition 5.1, which comprises a refined version of the scheme (1.9). Namely, an estimate of the form
Also (1.10) can be iterated, where now both the differentiability \(\vartheta \) and the integrability \(\beta \) change. The result is that
again uniformly in time. The last part of the paper, where we obtain the regularity in time, is quite standard for this kind of diffusion equations (see for example [10, page 118]). It amounts to using the already established spatial regularity and the information given by the equation. However, due to the fractional character of the spatial part of our equation, some care is needed in order to properly handle the time regularity. In particular, we have to treat the cases
separately. This is done in Proposition 6.2 and it yields the \(\gamma -\)Hölder continuity in time for any
given that the solution is \(\delta -\)Hölder continuous in the x variable. In particular, by the possible choice of \(\delta \), this yields that we may choose any \(\gamma <\Gamma (s,p)\), where the latter exponent is the one defined in (1.5).
1.5 Plan of the paper
The plan of the paper is as follows. In Sect. 2, we introduce the expedient spaces and notation used in this paper. In Sect. 3, we define local weak solutions and justify that we can insert certain test functions in the differentiated equation (see Lemma 3.3 below). This is followed by Sect. 4, where we prove that weak solutions are almost \(s-\)Hölder continuous in the spatial variable. In Sect. 5, we improve this result up to the exponent \(\Theta (s,p)\) defined in (1.5). This result is then used in Sect. 6, where we prove the corresponding Hölder regularity in time. Finally, in Sect. 7 we prove our main theorem.
The paper is complemented by an appendix, where for completeness we prove existence and uniqueness of weak solutions for the initial boundary value problem related to our equation. A comparison principle is also presented.
2 Preliminaries
2.1 Notation
We denote by \(B_r(x_0)\) the \(N-\)dimensional open ball of radius r centered at the point \(x_0\). The ball of radius r centered at the origin is denoted by \(B_r\). Its Lebesgue measure is given by
We use the following notation for the parabolic cylinder
Again, when \(x_0=0\) and \(t_0=0\), we simply write \(Q_{R,r}\).
Let \(1<p<\infty \), we denote by \(p'=p/(p-1)\) the conjugate exponent of p. For every \(\beta > 1\), we define the monotone function \(J_\beta :{\mathbb {R}}\rightarrow {\mathbb {R}}\) by
For a function \(\psi :{\mathbb {R}}^N\times {\mathbb {R}} \rightarrow {\mathbb {R}}\) and a vector \(h\in {\mathbb {R}}^N\), we define
and
It is not difficult to see that the following discrete Leibniz rule holds
2.2 Sobolev spaces
We now recall the main notations and definitions for the relevant fractional Sobolev–type spaces throughout the paper.
Let \(1\le q<\infty \) and let \(\psi \in L^q({\mathbb {R}}^N)\), for \(0<\beta \le 1\) we set
and for \(0<\beta <2\)
We then introduce the two Besov-type spaces
and
We also need the Sobolev-Slobodeckiĭ space
where the seminorm \([\,\cdot \,]_{W^{\beta ,q}({\mathbb {R}}^N)}\) is defined by
We endow these spaces with the norms
and
A few times we will also work with the space \(W^{\beta ,q}(\Omega )\) for a subset \(\Omega \subset {\mathbb {R}}^N\),
where we define
The space \(W_0^{\beta ,q}(\Omega )\) is the subspace of \(W^{\beta ,q}({\mathbb {R}}^N)\) consisting of functions that are identically zero in the complement of \(\Omega \).
2.3 Parabolic Banach spaces
Let \(I\subset {\mathbb {R}}\) be an interval and let V be a separable, reflexive Banach space, endowed with a norm \(\Vert \cdot \Vert _V\). We denote by \(V^*\) its topological dual space. Let us suppose that v is a mapping such that for almost every \(t\in I\), v(t) belongs to V. If the function \(t\mapsto \Vert v(t)\Vert _V\) is measurable on I and \(1\le p\le \infty \), then v is an element of the Banach space \(L^p(I;V)\) if and only if
By [28, Theorem 1.5], the dual space of \(L^p(I;V)\) can be characterized according to
We write \(v\in C(I;V)\) if the mapping \(t\mapsto v(t)\) is continuous with respect to the norm on V. We say that u is locally \(\alpha -\)Hölder continuous in space (respectively, locally \(\beta -\)Hölder continuous in time) on \(\Omega \times I\) and write
if for any compact set \(K\times J\subset \Omega \times I\),
That is, if \(u\in C^\alpha _{x}(K\times J)\) (respectively, \(u\in C^\beta _{t}(K\times J)\)).
2.4 Tail spaces
We recall the definition of tail space
which is endowed with the norm
For every \(x_0\in {\mathbb {R}}^N\), \(R>0\) and \(u\in L^q_{\alpha }({\mathbb {R}}^N)\), the following quantity
plays an important role in regularity estimates for solutions of fractional problems. We recall the following result, see for example [4, Lemmas 2.1 & 2.2] for the proof.
Lemma 2.1
Let \(\alpha >0\) and \(1\le q<m<\infty \). Then:
-
we have the continuous inclusion
$$\begin{aligned} L^{m}_{\alpha }({\mathbb {R}}^N)\subset L^{q}_{\alpha }({\mathbb {R}}^N); \end{aligned}$$ -
for every \(0<r<R\) and \(x_0\in {\mathbb {R}}^N\) we have
$$\begin{aligned} R^\alpha \,\sup _{x\in B_r(x_0)}\int _{{\mathbb {R}}^N{\setminus } B_R(x_0)} \frac{|u(y)|^q}{|x-y|^{N+\alpha }}\,\mathrm{d}y\le \left( \frac{R}{R-r}\right) ^{N+\alpha }\,\mathrm {Tail}_{q,\alpha }(u;x_0,R)^q. \end{aligned}$$
3 Weak formulation
3.1 Local weak solutions
In the following, we assume that \(\Omega \subset {\mathbb {R}}^N\) is a bounded open set in \({\mathbb {R}}^N\).
Definition 3.1
For any \(t_0,t_1\in {\mathbb {R}}\) with \(t_0<t_1\), we define \(I=(t_0,t_1]\). Let
We say that u is a local weak solution to the equation
if for any closed interval \(J=[T_0,T_1]\subset I\), the function u is such that
and it satisfies
for any \(\phi \in L^p(J;W^{s,p}(\Omega ))\cap C^1(J;L^2(\Omega ))\) which has spatial support compactly contained in \(\Omega \). In Eq. (3.2), the symbol \(\langle \cdot ,\cdot \rangle \) stands for the duality pairing between \(W^{s,p}(\Omega )\) and its dual space \((W^{s,p}(\Omega ))^*\).
We also say that u is a local weak subsolution if instead of the equality above, we have the \(\le \) sign, for any non-negative \(\phi \) as above. A local weak supersolution is defined similarly.
Remark 3.2
We observe that \(L^\infty ({\mathbb {R}}^N)\subset L^{p-1}_{s\,p}({\mathbb {R}}^N)\). This in turn implies that
We will use this fact repeatedly.
3.2 Regularization of test functions
Let \(\zeta :{\mathbb {R}}\mapsto {\mathbb {R}}\) be a nonnegative, even smooth function with compact support in \((-1/2,1/2)\), satisfying \(\int _{{\mathbb {R}}}\zeta (\tau )\,d\tau =1\). If \(g\in L^1((a,b))\), we define the convolution
where \(0<\varepsilon <\min \{b-t,\,t-a\}\). The following result justifies that we may take powers of differential quotients of a solution, as test functions. This is needed in the sequel. Here and in the rest of the paper, we will use the abbreviated notation
Lemma 3.3
(Discrete differentiation of the equation) Assume that u is a local weak solution of (3.1) with \(f=0\) in \(B_2\times (-2,0]\), such that
Let \(\eta \) be a non-negative Lipschitz function, with compact support in \(B_2\). Let \(\tau \) be a smooth non-negative function such that \(0\le \tau \le 1\) and
for some \(-1<T_0<T_1< 0\).
Then, for any locally Lipschitz function \(F:{\mathbb {R}}\rightarrow {\mathbb {R}}\) and any \(h\in {\mathbb {R}}^N\) such that \(0<|h|<\mathrm {dist\,}(\mathrm {supp\,}\eta , \partial B_2)/4\), we have
where \({\mathcal {F}}(t)=\int _0^t F(\rho )\,d\rho \).
Proof
Let \(\phi \in L^p((-1,0);W^{s,p}(B_2))\cap C^1((-1,0);L^2(B_2))\), whose spatial support is compactly contained in \(B_2\), uniformly in time. This means that
We then fix \(J=[T_0,T_1]\subset (-1,0)\). We want to use the time-regularization \(\phi ^\varepsilon \) as test function in (3.1). For this, we take
Then, we preliminary observe that from elementary properties of convolutions, Fubini’s Theorem and integration by parts, we have
For simplicity, we have set
Thus from (3.2) it follows that for \(0<\varepsilon <\varepsilon _0\)
Before proceeding further, we observe that by using an integration by parts, the term \(\Sigma (\varepsilon )\) can be rewritten as
where we also used that \(\zeta \) has compact support in \((-1/2,1/2)\). By further using a suitable change of variables, we can also write
By testing (3.6) with \(\phi _{-h}(x,t)=\phi (x-h,t)\) for \(0<|h|<h_0/4\) (recall the definition (3.5) of \(h_0\)), and then changing variables, we get
The quantity \(\Sigma _h(\varepsilon )\) is defined as in (3.7), with \(u_h\) in place of u. We subtract (3.6) from (3.8), so to get
for every \(\phi \in L^p((-1,0);W^{s,p}(B_2))\cap C^1((-1,0);L^2(B_2))\), whose spatial support satisfies (3.5). We take F as in the statement and use (3.9) with the test function
where
and \(\eta \) and \(\tau \) are as in the statement. By observing that
we get
Observe that we used the properties of \(\tau _\varepsilon \). In order to deal with the integral containing the time derivative of \(\delta _h u^\varepsilon \), we first observe that
since \({\mathcal {F}}(t)=\int _0^t F(\rho )\,d\rho \). Thus we can use an integration by parts, which yields
By inserting this into (3.10), we get
We recall that this is valid for
Before taking the limit as \(\varepsilon \) goes to 0, we first observe that for \(t\in [T_0-\varepsilon /2,T_1+\varepsilon /2]\) and \(x\in B_{2-2\,h}\) we have
This shows that we have the uniform \(L^\infty \) estimate
Finally, we pass to the limit in (3.11) as \(\varepsilon \) goes to 0. We start from the right-hand side: by using the local Lipschitz regularity of F and (3.12), we have
where \(C>0\) does not depend on \(\varepsilon \). Thus, by using that \(h_0=\mathrm {dist}(\mathrm {supp\,}\eta ,\partial B_2)\) and that \(0<|h|<h_0/4\), we get from the last estimate (after a change of variable)
The constant C is still independent of \(0<\varepsilon <\varepsilon _0\). If we now use that \(u\in C((-2,0];L^2_{\mathrm{loc}}(B_2))\), we get that the last quantity converges to 0, as \(\varepsilon \) goes to 0.
For the term
we proceed similarly as above. We observe that for \(0<|h|<h_0/4\), by using the local Lipschitz regularity of F and (3.12), we get
We can now use again that \(u\in C((-2,0];L^2_{\mathrm{loc}}(B_2))\) and obtain that the last quantity converges to 0, as \(\varepsilon \) goes to 0.
As for the term
we can proceed exactly as before, we omit the details. In a similar fashion, we can also show that
This is still similar to the previous limits. It is sufficient to use the expression (3.7), the uniform \(L^\infty \) estimate (3.12) and the fact \(u\in C((-2,0];L^2_{\mathrm{loc}}(B_2))]\), in order to apply the Lebesgue Dominated Convergence Theorem.
Finally, the convergence of the double integral requires quite lengthy computations and thus we prefer to postpone them to Appendix B below. \(\square \)
Remark 3.4
We observe that the global \(L^\infty \) bound on the weak solution is not needed in the previous result. It is sufficient to know that the weak solution is locally bounded. We refer to [32, Theorem 1.1] for local boundedness of weak solutions.
4 Spatial almost \(C^s\)-regularity
The following result is an integrability gain for the discrete derivative of order s of a local weak solution. This is the parabolic counterpart of [4, Proposition 4.1], to which we refer for all the missing details.
Proposition 4.1
Assume \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of \(u_t+(-\Delta _p)^s u=0\) in \(B_2\times (-2,0]\). We assume that
and that, for some \(q\ge p\) and \(0<h_0<1/10\), we have
for a radius \(4\,h_0<R\le 1-5\,h_0\) and two time instants \(-1<T_0<T_1\le 0\). Then we have
for every \(0<\mu <T_1-T_0\). Here \(C=C(N,s,p,q,h_0,\mu )>0\) and \(C\nearrow +\infty \) as \(h_0\searrow 0\) or \(\mu \searrow 0\).
Proof
We divide the proof into seven steps.
Step 1: Discrete differentiation of the equation. We take for the moment \(T_1<0\), then we will show at the end of the proof how to include the case \(T_1=0\). We already introduced the notation
For notational simplicity, we also set
Let \(\beta \ge 2\) and \(\vartheta \in {\mathbb {R}}\) be such that \(0<1+\vartheta \,\beta <\beta \), and use (3.4) for \(0<|h|< h_0\), where:
-
\(F(t)=J_{\beta +1}(t)=|t|^{\beta -1}\,t\), which is locally Lipschitz for \(\beta \ge 1\);
-
\(\eta \) is a non-negative standard Lipschitz cut-off function supported in \(B_{(R+r)/2}\), such that
$$\begin{aligned} \eta \equiv 1 \quad \text{ on } B_r\qquad \text{ and } \qquad |\nabla \eta |\le \frac{C}{R-r}=\frac{C}{4\,h_0}; \end{aligned}$$ -
\(\tau \) is a smooth function such that \(0\le \tau \le 1\) and
$$\begin{aligned} \tau \equiv 1\quad \text{ on } [T_0+\mu ,+\infty ), \qquad \tau \equiv 0\quad \text{ on } (-\infty ,T_0],\qquad |\tau '|\le \frac{C}{\mu }. \end{aligned}$$Here \(\mu \) is as in the statement, i.e. any positive number such that \(\mu <T_1-T_0\).
Note that the assumptions on \(\eta \) imply
After dividing by \(|h|^{1+\vartheta \,\beta }\), we obtain from Lemma 3.3,
The triple integral is now divided into three pieces:
where
and
where we used that \(\eta \) vanishes identically outside \(B_{(R+r)/2}\). We also suppressed the \(t-\)dependence inside the integrals, for notational simplicity. We also have the term in the right-hand side
By proceeding exactly as in Step 1 of the proof of [4, Proposition 4.1], we get the following lower bound for \({\mathcal {I}}_1(t)\)
where \(c=c(p,\beta )>0\) and \(C=C(p,\beta )>0\). We use that
and the estimate for \({\mathcal {I}}_1(t)\). This entails that
where we set \(\mathcal {{\widetilde{I}}}_{11}=\int _{T_0}^{T_1} {\mathcal {I}}_{11}\, \tau \, \mathrm{d}t\), \(\mathcal {{\widetilde{I}}}_{12}=\int _{T_0}^{T_1} {\mathcal {I}}_{12}\, \tau \, \mathrm{d}t\) and
and
Step 2: Estimates of the local terms \(\widetilde{{\mathcal {I}}}_{11}\) and \(\widetilde{{\mathcal {I}}}_{12}\). Here we can follow the same computations as in Step 2 of the proof of [4, Proposition 4.1], so to get
and
for some \(C=C(N,h_0,p,s,q)>0\). If we now use these estimates in (4.2), we get
with \(C=C(h_0,N,p,s,q,\beta )>0\).
Step 3: Estimates of the nonlocal terms \(\widetilde{{\mathcal {I}}}_2\) and \(\widetilde{{\mathcal {I}}}_3\). These two terms can be both treated in the same way. We only estimate \(\widetilde{{\mathcal {I}}}_2\) for simplicity. We can use that \(|u|\le 1\) on \({\mathbb {R}}^N\times [-1,0]\) to infer that
where \(C=C(p)>0\). We observe that for \(x\in B_{(R+r)/2}\) we have \(B_{(R-r)/2}(x)\subset B_{R}\). This entails
Hence, we obtain
by Young’s inequality. Here \(C=C(h_0,N,s,p,q,\beta )>0\) as before.
Step 4: Estimates of \({\mathcal {I}}_4\). By using that \(|u|\le 1\) in \({\mathbb {R}}^N\times [-1,0]\) and the properties of \(\tau \), we get
In the last inequality we further used Young’s inequality. By inserting the estimates (4.6) and (4.7) in (4.5), using that \(\tau \) is non-negative and such that \(\tau =1\) on \([T_0+\mu ,T_1]\), we obtain
This is the parabolic counterpart of [4, equation (4.10)]. Observe that the constant C now depends on \(\mu \), as well, and it blows-up as \(\mu \searrow 0\).
Step 5: Going back to the equation. In this step, we can simply reproduce Step 4 of the proof of [4, Proposition 4.1], so to obtain for any \(0< |\xi |,|h| < h_0\)
with \(C=C(N,h_0,s,\beta )>0\). This is the analogous of [4, equation (4.15)]. We then choose \(\xi =h\), take the supremum over h for \(0<|h|< h_0\) and integrate in time. Then (4.9) together with (4.8) imply
where \(C=C(N,h_0,p,q,s,\beta ,\mu )>0\). Since \((1+\vartheta \,\beta )/\beta < 1\), we can replace the first-order difference quotients in the right-hand side of (4.10) with second order ones, just by using [4, Lemma 2.6]. This gives
for some constant \(C=C(N,h_0,p,q,s,\beta ,\mu )>0\).
Step 6: Conclusion for \(T_1<0\). As in the final step of the step of [4, Proposition 4.1], we now fix
where \(q\ge p\) is as in the statement. These choices assure that
and
Then (4.11) becomes
where \(C=C(N,h_0,p,q,s)>0\). Up to a suitable modification of the constant C, we obtain in particular
as desired. Observe that we also used that \(r=R-4\,h_0\).
Step 7: Conclusion for \(T_1=0\). In this case, the previous proof does not directly work because it relies on Lemma 3.3, which needed \(T_1<0\). However, the constant C in (4.1) does not depend on \(T_1\), we can thus use a limit argument. By assumption, we have that for some \(q\ge p\) and \(0<h_0<1/10\), it holds
for a radius \(4\,h_0<R\le 1-5\,h_0\) and a time instant \(-1<T_0<0\). We fix \(0<\mu <-T_0\), then for every \(T<0\) such that \(\mu +T_0<T\) we have from Step 6
We then observe that
by the monotone convergence theorem. As for the second term on the left-hand side, we know by definition of local weak solution that
is a continuous function on \((-2,0]\), with values in \(L^2(B_{R-4\,h_0})\), for every fixed \(0<|h|<h_0\). Thus
This in turn implies thatFootnote 1
for every \(0<|h|<h_0\). By using (4.13) and (4.14) in (4.12), we get the desired conclusion for \(T_1=0\), as well. \(\square \)
As in [4, Theorem 4.2], by iterating the previous result, we can obtain the following regularity estimate.
Theorem 4.2
(Spatial almost \(C^s\) regularity) Let \(\Omega \subset {\mathbb {R}}^N\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that \(u\in L^\infty _{\mathrm{loc}}(I;L^\infty ({\mathbb {R}}^N))\). Then \(u\in C^\delta _{x,\mathrm loc}(\Omega \times I)\) for every \(0<\delta <s\).
More precisely, for every \(0<\delta <s\), \(R>0\) and every \((x_0,T_0)\) such that
there exists a constant \(C=C(N,s,p,\delta )>0\) such that
Proof
We assume for simplicity that \(x_0=0\) and \(T_0=0\), then we set
Let \(\alpha \in [-R^{s\,p}(1-{\mathcal {M}}_R^{2-p}),0]\) and set
By taking into account the scaling properties of our equation (see Remark 1.1), the function \(u_{R,\alpha }\) is a local weak solution of
and satisfies
We will prove that \(u_{R,\alpha }\) satisfies the estimate
for \(C=C(N,s,p,\delta )>0\) independent of \(\alpha \). By scaling back, this would give
Since \(\alpha \in [-R^{s\,p}(1-{\mathcal {M}}_R^{2-p}),0]\) and \({\mathcal {M}}_R^{2-p}\le 1\), this in turn would imply
which is the desired result. In what follows, we suppress the subscript \({R,\alpha }\) and simply write u in place of \(u_{R,\alpha }\), in order not to overburden the presentation.
We fix \(0<\delta <s\) and choose \(i_\infty \in {\mathbb {N}}{\setminus }\{0\}\) such that
Then we define the sequence of exponents
We define also
We note that
By applying Proposition 4.1 (ignoring the second term in the left-hand side of (4.1)) withFootnote 2
and
and observing that \(R_i-4\,h_0=R_{i+1}+4\,h_0\), we obtain the iterative scheme of inequalities:
-
for \(i=0\)
$$\begin{aligned} \int _{-(R_1+4h_0)}^0 \sup \limits _{0<|h|< h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{s}}\right\| _{L^{q_1}(B_{R_1+4h_0})}^{q_1}\mathrm{d}t \le C\,\displaystyle \int _{-\frac{7}{8}}^0 \sup \limits _{0<|h|< h_0}\left( \left\| \dfrac{\delta ^2_h u }{|h|^s}\right\| _{L^p(B_{7/8})}^p+1\right) \mathrm{d}t\\ \end{aligned}$$ -
for \(i=1,\dots ,i_\infty -2\)
$$\begin{aligned}&\int _{-(R_{i+1}+4h_0)}^0 \sup \limits _{0<|h|< h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{s}}\right\| _{L^{q_i+1}(B_{R_{i+1}+4h_0})}^{q_i+1}\mathrm{d}t \\&\quad \le C\,\int _{-(R_{i}+4h_0)}^0\sup \limits _{0<|h|< h_0}\left( \left\| \dfrac{\delta ^2_h u }{|h|^s}\right\| _{L^{q_i}(B_{R_i+4h_0})}^{q_i}+1\right) \mathrm{d}t, \end{aligned}$$ -
finally, for \(i=i_\infty -1\)
$$\begin{aligned} \begin{aligned}&\int _{-\frac{3}{4}}^0\sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{s}}\right\| _{L^{q_{i_\infty }}(B_{3/4})}^{q_{i_\infty }}\mathrm{d}t\\&\quad =\int _{-(R_{i_\infty }+4h_0)}^0\sup _{0<|h|< h_0}\left\| \frac{\delta ^2_h u}{|h|^{s}}\right\| _{L^{p+i_\infty }(B_{R_{i_\infty }+4h_0})}^{p+i_\infty }\mathrm{d}t \\&\quad \le C\,\int _{-(R_{i_\infty -1}+4h_0)}^0\sup _{0<|h|< h_0}\left( \left\| \frac{\delta ^2_h u }{|h|^s}\right\| _{L^{p+i_\infty -1}(B_{R_{i_\infty -1}+4h_0})}^{p+i_\infty -1}+1\right) \,\mathrm{d}t. \end{aligned} \end{aligned}$$
Here \(C=C(N,\delta ,p,s)>0\) as always. We note that by using the relation
and then appealing to [3, Proposition 2.6], we have
where we also have used the assumptions (4.16) on u. Hence, the iterative scheme of inequalities leads us to
It is now time to exploit the full power of Proposition 4.1: we apply it once more, with
We obtain (ignoring the first term in the left-hand side of (4.1), this time)
Since this is valid for every \(-1/2\le T_1\le 0\), this in turn implies that
Take now \(\chi \in C_0^\infty (B_{9/16})\) such that
In particular, we have for all \(0<|h|<h_0\)
We also recall that
Hence, for \(0<|h|< h_0\) and any \(t\in [-5/8,0]\)
by (4.18). Finally, by noting that thanks to the choice of \(i_\infty \) we have
we may invoke the Morrey-type embedding of [4, Theorem 2.8] with
Thus we obtain
for any \(t\in [-1/2,0]\), where we used (4.19). This concludes the proof. \(\square \)
Remark 4.3
Under the assumptions of the previous theorem, a covering argument combined with (4.15) implies the following more flexible estimate: for every \(0<\sigma <7/8\)
with C now depending on \(\sigma \) as well (and blowing-up as \(\sigma \nearrow 7/8\)). Indeed, if \(\sigma \le 1/2\) then this is immediate. If \(1/2<\sigma <7/8\), then we can cover \(Q_{\sigma R,\sigma R^{s\,p}}(x_0,T_0)\) with a finite number of cylinders
where
and \(r=R/C_{\sigma ,s,p}>0\) is a suitable radius, such that
and
By using (4.15) on each of these cylinders and the fact that \(r=R/C_{\sigma ,s,p}\), we get
By taking the supremum over \(1\le i\le k\) and \(1\le j\le m\), we get the desired conclusion.
5 Improved spatial Hölder regularity
Once we know that solutions are locally spatially \(\delta -\)Hölder continuous for any \(0<\delta <s\), we can obtain the following improvement of Proposition 4.1. The latter provided a recursive gain of integrability. In contrast, the next result provides a gain which is interlinked between differentiability and integrability.
Proposition 5.1
Assume \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of \(u_t+(-\Delta _p)^s u=0\) in \(B_2\times (-2,0]\), such that
Assume further that for some \(0<h_0<1/10\) and \(\vartheta <1\), \(\beta \ge 2\) such that \((1+\vartheta \, \beta )/\beta <1\), we have
for a radius \(4\,h_0<R\le 1-5\,h_0\) and two time instants \(-3/4\le T_0< T_1\le 0\). Then it holds
for every \(0<\mu <T_1-T_0\). Here C depends on N, \(h_0\), s, p, \(\mu \) and \(\beta \).
Proof
This is analogous to the proof of [4, Proposition 5.1]. As above, we will refer to [4] for the main computations and only list the major changes.
We first notice that it is sufficient to prove (5.1) for \(T_1<0\), with a constant independent of \(T_1\). Then the same argument of Step 7 in Proposition 4.1 will be enough to handle the case \(T_1=0\), as well.
We go back to the estimates in the proof of Proposition 4.1. The acquired knowledge on the spatial regularity of u permits to improve the estimate on the term \({\mathcal {I}}_{11}(t)\) defined in (4.3). From Theorem 4.2 and Remark 4.3, we can choose
such that
Using this together with the assumed regularity of \(\eta \), we have for \(x,y\in B_{R}\) and \(t\in [T_0,T_1]\)
Thanks to the choice of \(\varepsilon \), the last exponent is strictly larger than \(-N\) and we may conclude
for any \(x\in B_R\). A similar estimate holds for the other term of \({\mathcal {I}}_{11}(t)\) containing \(|u_h(x,t)-u_h(y,t)|\). Therefore, by suppressing as before the \(t-\)dependence for simplicity, we have the estimate
As for \({\mathcal {I}}_{12}\), by going back to its definition (4.4) and using the properties of the cut-off function \(\eta \), we get
where we used the local \(L^\infty \) bound on u, as above. In addition, from the first inequality in (4.6) together with the properties of the cut-off function \(\tau \), we have
By combining these new estimates with (4.7) and (4.2), we can reproduce the last part of [4, Proposition 5.1] and arrive at
for some \(C=C(N,h_0,p,s,\beta )>0\). By appealing again to [4, Lemma 2.6] and using that
we may replace the first-order differential quotients in the right-hand side by second order ones. This leads to
for some \(C=C(N,h_0,p,s,\beta )>0\). By recalling again that \(r=R-4\,h_0\), we eventually conclude the proof. \(\square \)
We are now ready to prove the claimed Hölder regularity in space.
Theorem 5.2
Let \(\Omega \) be a bounded and open set, let \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Suppose u is a local weak solution of
such that \(u\in L^\infty _{\mathrm{loc}}(I;L^\infty ({\mathbb {R}}^N))\). Then \(u\in C^\delta _{x,\mathrm loc}(\Omega \times I)\) for every \(0<\delta <\Theta (s,p)\), where \(\Theta (s,p)\) is defined in (1.5).
More precisely, for every \(0<\delta <\Theta (s,p)\), \(R>0\), \(x_0\in \Omega \) and \(T_0\) such that
there exists a constant \(C=C(N,s,p,\delta )>0\) such that
Proof
By the same scaling argument as in the proof of Theorem 4.2, it is enough to prove that
under the assumption that u is a local weak solution of
which satisfies (4.16). Define for \(i\in {\mathbb {N}}\), the sequences of exponents
and
By induction, we see that \(\{\vartheta _i\}_{i\in {\mathbb {N}}}\) is explicitely given by the increasing sequence
and thus
The proof is now split into two different cases.
Case 1: \(s\,p\le (p-1)\). Fix \(0<\delta <s\,p/(p-1)\) and choose \(i_\infty \in {\mathbb {N}}{\setminus }\{0\}\) such that
This is feasible, since
Define also
We note that
By applyingFootnote 3 Proposition 5.1 (ignoring the second term of the left-hand side of (5.1)) with
and
and observing that \(R_i-4\,h_0=R_{i+1}+4\,h_0\), that \(T_0^{i+1}=T_0^i+\mu \) and by construction
we obtain the iterative scheme of inequalities:
-
for \(i=0\)
$$\begin{aligned} \int _{T_0^1}^0\sup \limits _{0<|h|< h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{\frac{1+\vartheta _1\beta _1}{\beta _1}}}\right\| _{L^{\beta _1} (B_{R_1+4h_0})}^{\beta _1}\mathrm{d}t \le C\,\displaystyle \int _{-\frac{3}{4}}^0\sup \limits _{0<|h|< h_0}\left( \left\| \dfrac{\delta ^2_h u }{|h|^s}\right\| _{L^{p}(B_{7/8})}^p+1\right) \mathrm{d}t; \end{aligned}$$ -
for \(i=1,\ldots ,i_\infty -2\)
$$\begin{aligned} \begin{aligned}&\int _{T_0^{i+1}}^0\sup \limits _{0<|h|< h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{\frac{1+\vartheta _{i+1}\beta _{i+1}}{\beta _{i+1}}}}\right\| _{L^{\beta _{i+1}}(B_{R_{i+1}+4h_0})}^{\beta _{i+1}}\mathrm{d}t\\&\quad \le C\,\displaystyle \int _{T_0^{i}}^0\sup \limits _{0<|h|< h_0}\left( \left\| \dfrac{\delta ^2_h u }{|h|^{\frac{1+\vartheta _{i}\beta _{i}}{\beta _{i}}}}\right\| _{L^{\beta _i}(B_{R_i+4\,h_0})}^{\beta _i}+1\right) \mathrm{d}t ; \end{aligned} \end{aligned}$$ -
finally, for \(i=i_\infty -1\)
$$\begin{aligned} \begin{aligned}&\displaystyle \int _{-\frac{5}{8}}^0\sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{\frac{1}{\beta _{i_\infty }}+\vartheta _{i_\infty }}}\right\| _{L^{\beta _{i_\infty }}(B_{3/4})}^{\beta _{i_\infty }}\mathrm{d}t\\&\quad \le \displaystyle C\int _{T_0^{i_\infty -1}}^0\sup _{0<|h|< h_0}\left( \left\| \frac{\delta ^2_h u }{|h|^{\frac{1+\vartheta _{i_\infty -1}\beta _{i_\infty -1}}{\beta _{i_\infty -1}}}}\right\| _{L^{\beta _{i_\infty -1}}(B_{R_{i_\infty -1}+4\,h_0})}^{\beta _{i_\infty -1}}+1\right) \mathrm{d}t. \end{aligned} \end{aligned}$$
Here \(C=C(N,p,s,\delta )>0\) as always. As in (4.17) we have
Hence, the previous iterative scheme of inequalities implies
Now we apply Proposition 5.1 once more, this time with
We obtain (now ignoring the first term in the left-hand side of (5.1))
Since this is valid for every \(-1/2\le T_1\le 0\), we obtain
From here, we may repeat the arguments at the end of the proof of Theorem 4.2 (see (4.19)) and use the Morrey–type embedding of [4, Theorem 2.8], with
to obtain
which concludes the proof in this case.
Case 2: \(s\,p> (p-1)\). Fix \(0<\delta <1\). Let \(i_\infty \in {\mathbb {N}}{\setminus }\{0\}\) be such that
Observe that such a choice is feasible, since
Now choose \(j_\infty \) so that
and let
Define also
We note that
By applyingFootnote 4 Proposition 5.1 with
and
and observing that \(R_i-4\,h_0=R_{i+1}+4\,h_0\), that \(T_0^{i+1}=T_0^i+\mu \) and that
we arrive as in Case 1 at
since \(\gamma <1\le 1/\beta _{i_\infty }+\vartheta _{i_\infty }\). We now apply Proposition 5.1 with
Observe that by construction we have
and using that \(s\,p>(p-1)\)
This gives the following inequalities:
-
for \(i=i_\infty ,\ldots ,i_\infty +j_\infty -2\)
$$\begin{aligned} \int _{T_0^{i+1}}^0\sup \limits _{|h|\le h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{\gamma }}\right\| _{L^{\beta _{i+1}}(B_{R_{i+1}+4h_0})}^{\beta _{i+1}}\mathrm{d}t\le C\,\int _{T_0^i}^0\sup \limits _{0<|h|< h_0}\left( \left\| \dfrac{\delta ^2_h u }{|h|^{\gamma }}\right\| _{L^{\beta _i}(B_{R_i+4h_0})}^{\beta _i}+1\right) \mathrm{d}t, \end{aligned}$$ -
for \(i= i_\infty +j_\infty -1\)
$$\begin{aligned} \begin{aligned} \int _{-\frac{5}{8}}^0&\sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{\gamma }} \right\| _{L^{\beta _{i_\infty +j_\infty }}(B_{3/4})}^{\beta _{i_\infty +j_\infty }}\mathrm{d}t\\&\le C\int _{T_0^{i_\infty +j_\infty -1}}^0\sup _{0<|h|< h_0}\left( \left\| \frac{\delta ^2_h u }{|h|^{\gamma }}\right\| _{L^{\beta _{i_\infty +j_\infty -1}} (B_{R_{i_\infty +j_\infty -1}+4h_0})}^{\beta _{i_\infty +j_\infty -1}}+1\right) \mathrm{d}t. \end{aligned} \end{aligned}$$
Hence, recalling that \(\gamma =1-\varepsilon \), we conclude
Now we apply Proposition 5.1 again, with
We obtain (ignoring again the first term in the left-hand side)
Once we land here, as before we can repeat the arguments at the end of the proof of Theorem 4.2 and use the Morrey-type embedding, this time with
This gives
and the proof is concluded. \(\square \)
6 Regularity in time
In this section, we prove Hölder regularity in time using the previously obtained regularity in space. This approach uses energy estimates to control the growth of local integrals which yields a Campanato–type estimate. For \(u\in L^1(B_{R}(x_0))\), we will use the notation
When the center \(x_0\) is clear from the context, we often simply write \({\overline{u}}_R\). For \(u\in L^1(Q_{R,r}(x_0,t_0))\), we set
Again, when the center \((x_0,t_0)\) is clear from the context, we simply write \({\overline{u}}_{R,r}\).
The following simple Poincaré–type inequality will be useful.
Lemma 6.1
Let \(1\le p<\infty \) and let \(B_r = B_r(x_0)\). Suppose that \(u\in W^{s,p}(B_r)\), then for any nonnegative \(\eta \in C_0^\infty (B_r)\) such that \({\overline{\eta }}_r=1\), there holds
Proof
By using the fact that \(\int _{B_r}\eta \,\mathrm{d}x=|B_r|\) and Jensen’s inequality, we obtain
This concludes the proof. \(\square \)
Proposition 6.2
Let \(p\ge 2\) and suppose that u is a local weak solution of
such that
and
where \(\Theta (s,p)\) is the exponent defined in (1.5). Then there is a constant \(C= C(N,s,p,K_\delta ,\delta )>0\) such that
where
In particular, \(u\in C^\gamma _{t}(Q_{\frac{1}{4},\frac{1}{4}})\) for any \(\gamma <\Gamma (s,p)\), where \(\Gamma (s,p)\) is the exponent defined in (1.5).
Proof
We take \((x_0,t_0)\in Q_{1/4,1/4}\) and choose
Consider the parabolic cylinder
Observe that by construction we have
Let \(\eta \in C_0^\infty (B_{r/2}(x_0))\) be a non-negative cut-off function, such that
for some constant \(C=C(\Vert \eta \Vert _{L^\infty (B_{r/2}(x_0))},N)>0\). Observe that, thanks to the condition on its average, we have
Thus the constant appearing in (6.1) will only depend on N, s and p.
We now write
where we have set
Then
We first note that
Thus it suffices to estimate \(A_1\) and \(A_3\). In view of Lemma 6.1, we have
for some \(C=C(N,s,p)>0\). Recalling that \(\delta >s\) and using the spatial Hölder continuity of u, we find that
Indeed, by observing that for every \(x\in B_r(x_0)\) we have \(B_r(x_0)\subset B_{2\,r}(x)\subset B_{1/2}\), we get
where we used spherical coordinates to compute the last integral. Observe that the width \(\theta \) of the time interval does not come into play here.
We now turn to \(A_3\) and first note that
thus
If \(T_0,T_1 \in (t_0-\theta ,t_0]\) with \(T_0<T_1\), we use the weak formulation (3.2) with \(\phi (x,t)=\eta (x)\) and \(f=0\), to obtain
In order to control \(J_2\), we claim that for \(t\in [-1/2,0]\), \(x\in B_r(x_0)\) and \(y\in {\mathbb {R}}^N\),
Indeed, if \(y\in B_{1/2}\) this follows directly from the assumption. On the other hand, if \(y\in {\mathbb {R}}^N{\setminus } B_{1/2}\), then by construction
Additionally, if \(y\in {\mathbb {R}}^N{\setminus } B_r(x_0)\) and \(x\in B_{r/2}(x_0)\), we have
Thus, by using this and (6.7), we get
for some \(C = C(\delta ,N,s,p,K_\delta )>0\). Observe that we used that \(\delta \,(p-1)-s\,p<0\), in order to assure that the integral on \({\mathbb {R}}^N{\setminus } B_r(x_0)\) converges.
As for \(J_1\), we have for \(\delta >s\)
for some \(C=C(N,s,p,\delta )>0\). By recalling (6.5) and using the estimates on \(J_1\) and \(J_2\) in (6.6), we have thus shown that
Hence, by also using (6.4) and (6.3), we get
We now have to distinguish two cases:
\(\bullet \) Case \(s\,p\ge (p-1)\). We choose \(\theta \) as follows
Observe that since \(s\,p\ge (p-1)\), then \(\Theta (s,p)=1\) and we always haveFootnote 5
We thus obtain from (6.8)
By the characterization of Campanato spaces on \({\mathbb {R}}^{N+1}\) with respect to a general metric (see [11, Teorema 3.I] and also [16, Theorem 3.2]), this implies that u is \(\delta -\)Hölder continuous in \(Q_{1/4,1/4}\) with respect to the metric
By keeping (6.9) into account, we can infer that \({\widetilde{d}}\) is a true metric. Thus, in particular, we have the estimate
where \(C=C(\delta ,K_\delta , N,s,p)>0\). Observe that the continuous function
is increasing and that
Thus for every \(0<\gamma <1/(s\,p-(p-2))\), there exists \(s<\delta <1\) such that
The proof is over in this case.
\(\bullet \) Case \(s\,p< (p-1)\). In this case, we revert the hierarchy between time and space and choose r as follows
Observe that the exponent on \(\theta \) is positive: indeed, for \(p=2\) this is straightforward, while for \(p>2\) we use that
We further notice that now
up to choose \(\delta \) sufficiently closeFootnote 6 to \(s\,p/(p-1)\). This time, we obtain from (6.8)
Again by the Campanato–type theorem of [11, Teorema 3.I], this shows that u is \((\delta /(s\,p-(p-2)\,\delta ))-\)Hölder continuous in \(Q_{1/4,1/4}\) with respect to the metric
Observe that this is indeed a metric, thanks to (6.10). In particular, we have the estimate
where \(C=C(\delta ,K_\delta , N,s,p)>0\). We now use that the continuous function
is increasing and that
Thus, for every \(\gamma <1\), there exists \(s<\delta <s\,p/(p-1)\) such that
This concludes the proof in this case, as well. \(\square \)
7 Proof of the main theorem
Before proving our main result, we will need the following lemma, which allows us to control the parabolic Sobolev-Slobodeckiĭ seminorm of a local weak solution u in terms of its \(L^\infty \) norm.
Lemma 7.1
Let \(\Omega \subset {\mathbb {R}}^N\) be a bounded and open set, \(I=(t_0,t_1]\), \(p\ge 2\) and \(0<s<1\). Let u be a local weak solution of
such that
Then for every \(x_0\in \Omega \) and \(T_0\in I\) such that \(Q_{2\,R,2\,R^{s\,p}}(x_0,T_0)\Subset \Omega \times I\), we have
for some \(C=C(N,s,p)>0\).
Proof
Without loss of generality, we may suppose that \(x_0=0\) and \(T_0=0\). Let us set
Then \({\widetilde{u}}\) is still a local weak solution in \(\Omega \times I\) and \({\widetilde{u}}\ge 1\) in \({\mathbb {R}}^N\times [-R^{s\,p},0]\). For all \(\phi (x,t) = \eta (x)\, \psi (t)\) with
and \(\eta \in C_0^{\infty }(B_{2\,R})\), we get from a slight modification of [31, Lemma 2.2]
We choose \(\eta \) such that
and \(\psi \) such that
It is then a routine matter to show that
where \(C= C(N,s,p)>0\) and we used that \(p\ge 2\) and \(k\ge 1\). This proves the claimed estimate. \(\square \)
We are now in the position to prove Theorem 1.2.
Proof of Theorem 1.2
The continuity in space is contained in Theorem 5.2, thus we only need to prove the continuity in time. We take for simplicity \(T_0=0\). If u is a local weak solution as in the statement, we obtain from (5.2)
An application of Lemma 7.1 gives
We set
then for \(\alpha \in [-R^{s\,p}(1-{\mathcal {N}}_R^{2-p}),0]\), we define the rescaled function
This is a local weak solution in \(B_2(x_0)\times (-2,0]\) satisfying the hypothesis of Proposition 6.2. Indeed, by construction
and the estimate on the spatial Hölder seminorm (6.2) of \(u_{R,\alpha }\) follows from (7.1). From Proposition 6.2 we obtain
for every \(0<\gamma <\Gamma (s,p)\). The claimed result follows by scaling back and varying \(\alpha \) as in the proof of Theorem 4.2. \(\square \)
Notes
We use the following standard fact: if \(\{f_n\}_{n\in {\mathbb {N}}}\) converges to f in \(L^\alpha (E)\), then
$$\begin{aligned} \liminf _{n\rightarrow \infty } \Vert f_n\Vert _{L^\beta (E)}\ge \Vert f\Vert _{L^\beta (E)}, \end{aligned}$$for any \(\beta \not =\alpha \).
We observe that by construction we have
$$\begin{aligned} 4\,h_0<R_i\le 1-5\,h_0,\qquad \text{ for } i=0,\dots ,i_\infty -1. \end{aligned}$$Thus these choices are admissible in Proposition 4.1.
Note that in this case we will always have \(1+\vartheta _i\beta _i<\beta _i\), so that the proposition applies.
Note that for \(i\le i_\infty -1\) we have \(1+\vartheta _i\,\beta _i<\beta _i\), so that the proposition applies.
Indeed, observe that
$$\begin{aligned} s\,p\ge (p-1)=(p-2)+1>\delta \,(p-2)+1, \end{aligned}$$thanks to the fact that \(0<\delta <1\). This in turn implies (6.9).
More precisely, it is sufficient to take
$$\begin{aligned} \delta =\frac{s\,p}{p-1}-\varepsilon , \end{aligned}$$with \(0<\varepsilon <s/(p-1)\) such that
$$\begin{aligned} \varepsilon \,(p-2)\le 1-\frac{s\,p}{p-1}. \end{aligned}$$Such a choice is feasible, since now \(s\,p<(p-1)\).
With these identifications, we have \(V\subset H\subset V^*\). This is sometimes called in the literature Gelfand triple.
By construction, for \(x\in {\mathbb {R}}^N{\setminus }\Omega \) and \(t\in J\) we have
$$\begin{aligned} u^\varepsilon (x,t)=\frac{1}{\varepsilon }\,\int _{-\frac{\varepsilon }{2}}^\frac{\varepsilon }{2} \zeta \left( \frac{\sigma }{\varepsilon }\right) \,u(x,t-\sigma )\,d\sigma =\frac{1}{\varepsilon }\,\int _{-\frac{\varepsilon }{2}}^\frac{\varepsilon }{2} \zeta \left( \frac{\sigma }{\varepsilon }\right) \,g(x,t-\sigma )\,d\sigma \le M, \end{aligned}$$for \(\varepsilon \ll 1\). Thus \((u^\varepsilon (\cdot ,t)-M)_+\in X^{s,p}_0(\Omega ,\Omega ')\).
References
B. Abdellaoui, A. Attar, R. Bentifour, I. Peral, On fractional \(p-\)Laplacian parabolic problem with general data, Ann. Mat. Pura Appl., 197 (2018), 329–356.
V. Bögelein, Global gradient bounds for the parabolic \(p-\)Laplacian system, Proc. Lond. Math. Soc. (3), 111 (2015), 633–680.
L. Brasco, E. Lindgren, Higher Sobolev regularity for the fractional \(p-\)Laplace equation in the superquadratic case, Adv. Math., 304 (2017), 300–354.
L. Brasco, E. Lindgren, A. Schikorra, Higher Hölder regularity for the fractional \(p-\)Laplacian in the superquadratic case, Adv. Math., 338 (2018), 782–846.
L. Brasco, E. Parini, The second eigenvalue of the fractional \(p-\)Laplacian, Adv. Calc. Var., 9 (2016), 323–355.
M. Cozzi, Regularity results and Harnack inequalities for minimizers and solutions of nonlocal problems: a unified approach via fractional De Giorgi classes, J. Funct. Anal., 272 (2017), 4762–4837.
L. A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903–1930.
H. Chang-Lara, G. Dávila, Regularity for solutions of non local parabolic equations, Calc. Var. Partial Differential Equations, 49 (2014), 139–172.
H. Chang-Lara, G. Dávila, Regularity for solutions of non local parabolic equations II, J. Differential Equations, 256 (2014), 130–156.
Y. Z. Chen, E. DiBenedetto, Boundary estimates for solutions of nonlinear degenerate parabolic systems, J. Reine Angew. Math., 395 (1989), 102–131.
G. Da Prato, Spazi \({\cal{L}}^{p,\theta }(\Omega ,\delta )\) e loro proprietà, Ann. Mat. Pura Appl. (4), 69 (1965), 383–392.
E. DiBenedetto, Degenerate parabolic equations, Springer, New York, (1993).
E. DiBenedetto, U. Gianazza, V. Vespri, Harnack’s inequality for degenerate and singular parabolic equations, Springer, New York, (2012).
A. Di Castro, T. Kuusi, G. Palatucci, Local behavior of fractional \(p-\)minimizers, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1279–1299.
A. Di Castro, T. Kuusi, G. Palatucci, Nonlocal Harnack inequalities, J. Funct. Anal., 267 (2014), 1807–1836.
P. Górka, Campanato theorem on metric measure spaces, Ann. Acad. Sci. Fenn. Math., 34 (2009), 523–528.
R. Hynd, E. Lindgren, Hölder estimates and large time behavior for a nonlocal doubly nonlinear evolution, Anal. PDE, 9 (2016), 1447–1482.
A. Iannizzotto, S. Mosconi, M. Squassina, Fine boundary regularity for the degenerate fractional \(p-\)Laplacian, J. Funct. Anal., 279 (2020), 108659
A. Iannizzotto, S. Mosconi, M. Squassina, Global Hölder regularity for the fractional \(p-\)Laplacian, Rev. Mat. Iberoam., 32 (2016), 1353–1392.
H. Ishii, G. Nakamura, A class of integral equations and approximation of \(p-\)Laplace equations, Calc. Var. Partial Differential Equations, 37 (2010), 485–522.
J. Korvenpää, T. Kuusi, G. Palatucci, The obstacle problem for nonlinear integro-differential operators, Calc. Var. Partial Differential Equations, 55 (2016), Paper No. 63, 29 pp.
T. Kuusi, G. Mingione, Y. Sire, Nonlocal equations with measure data, Comm. Math. Phys., 337 (2015), 1317–1368.
T. Kuusi, G. Mingione, Y. Sire, Nonlocal self-improving properties, Anal. PDE, 8 (2015), 57–114.
E. Lindgren, Hölder estimates for viscosity solutions of equations of fractional \(p-\)Laplace type, NoDEA Nonlinear Differential Equations Appl., 23 (2016), Art. 55, 18 pp.
J. M. Mazon, J. D. Rossi, J. Toledo, Fractional \(p-\)Laplacian evolution equations, J. Math. Pures. Appl., 105 (2016), 810–844.
D. Puhst, On the evolutionary fractional \(p-\)Laplacian, Appl. Math. Res. Express. AMRX, 2 (2015), 253–273.
A. Schikorra, Nonlinear commutators for the fractional \(p-\)Laplacian and applications, Math. Ann., 366 (2016), no. 1–2, 695–720.
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997.
L. Silvestre, Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient, Discrete Contin. Dyn. Syst., 28 (2010), 106–1081.
L. Silvestre, Hölder estimates for advection fractional-diffusion equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 11 (2012), 843–855.
M. Strömqvist, Local boundedness of solutions to nonlocal parabolic equations modeled on the fractional \(p-\)Laplacian, J. Differential Equations, 266 (2019), 7948–7979.
M. Strömqvist, Harnack’s inequality for parabolic nonlocal equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 36 (2019), 1709–1745.
J. L. Vázquez, The Dirichlet problem for the fractional \(p-\)Laplacian evolution equation, J. Differential Equations, 260 (2016), 6038–6056.
J. L. Vázquez, The evolution fractional \(p-\)Laplacian equation in \({\mathbb{R}}^N\). Fundamental solution and asymptotic behaviour, Nonlinear Anal., 199 (2020), 112034, 32 pp.
M. Warma, Local Lipschitz continuity of the inverse of the fractional \(p-\)Laplacian, Hölder type continuity and continuous dependence of solutions to associated parabolic equations on bounded domains, Nonlinear Anal., 135 (2016), 129–157.
Acknowledgements
We thank Eleonora Cinti for drawing our attention on the papers [8, 9]. E. L. is supported by the Swedish Research Council, grant no. 2012-3124 and 2017-03736. Part of this work has been done during a visit of L. B. to Uppsala and a visit of E. L. to Bologna and Ferrara. The paper has been finalized during the conference “Nonlinear averaging and PDEs ”, held in Levico Terme in June 2019. The hosting institutions and the organizers are kindly acknowledged.
Funding
Open access funding provided by Uppsala University.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix A. Existence for an initial boundary value problem
In order to give the definition of weak solution for an initial boundary value problem, we need to define a suitable functional space. We assume that \(\Omega \Subset \Omega '\subset {\mathbb {R}}^N\), where \(\Omega '\) is a bounded open set in \({\mathbb {R}}^N\). Given a function
we define as in [21] (see also [4, Proposition 2.12]) the space
When \(\psi \equiv 0\), the boundedness of \(\Omega '\) entails that
We endow the space \(X_0^{s,p}(\Omega ,\Omega ')\) with the norm \(W^{s,p}(\Omega ')\), then this is a reflexive Banach space. Thanks to the previous inclusion, we also have that
Definition A.1
Let \(I=[t_0,t_1]\) and \(p\ge 2\). With the notation above, assume that the functions \(u_0,f\) and g satisfy
We say that u is a weak solution of the initial boundary value problem
if the following properties are verified:
-
\(u\in L^p(I;W^{s,p}(\Omega '))\cap L^{p-1}(I;L_{s\,p}^{p-1}({\mathbb {R}}^N))\cap C(I;L^2(\Omega ))\);
-
\(u\in X^{s,p}_{{\mathbf {g}}(t)}(\Omega ,\Omega ')\) for almost every \(t\in I\), where \(({\mathbf {g}}(t))(x)=g(x,t)\);
-
\(\lim _{t\rightarrow t_0}\Vert u(\cdot ,t) - u_0\Vert _{L^2(\Omega )}=0\);
-
for every \(J=[T_0,T_1]\subset I\) and every \(\phi \in L^{p-1}(J;X_0^{s,p}(\Omega ,\Omega '))\cap C^1(J;L^2(\Omega ))\)
$$\begin{aligned} \begin{aligned}&-\int _J\int _\Omega u(x,t)\,\partial _t\phi (x,t)\,\mathrm{d}x\,\mathrm{d}t\\&\qquad + \int _J\iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N}\frac{J_p(u(x,t) -u(y,t))\,(\phi (x,t)-\phi (y,t))}{|x-y|^{N+s\,p}}\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}t \\&\quad = \int _\Omega u(x,T_0)\,\phi (x,T_0)\,\mathrm{d}x -\int _\Omega u(x,T_1)\,\phi (x,T_1)\,\mathrm{d}x \\&\qquad + \int _J\langle f(\cdot ,t),\phi (\cdot ,t)\rangle \,\mathrm{d}t. \end{aligned} \end{aligned}$$
The starting point for proving the existence of weak solutions is an abstract theorem for parabolic equations in Banach spaces. Before stating the theorem, we will briefly explain its framework. Let V be a separable reflexive Banach space and let H be a Hilbert space that we identify with its dual, i.e. \(H^* = H\). Suppose that V is dense and continuously embedded in H. If \(v\in V\) and \(h\in H\), we identify h as an element of \(V^*\) through the relationFootnote 7
Here \(\langle \cdot ,\cdot \rangle \) denotes the duality pairing between V and \(V^*\) and \((\cdot ,\cdot )_H\) denotes the scalar product in H. Let I be an interval and \(1<p<\infty \). By [28, Proposition 1.2, Chapter III], we have
and
More generally, by [28, Corollary 1.1, Chapter III], for every \(u,v\in W_p(I)\) the scalar product \(t\mapsto (u(t),v(t))_H\) is an absolutely continuous function and there holds
We recall that an operator \({\mathcal {A}}:V\rightarrow V^*\) is said to be
-
monotone if for every \(u,v\in V\),
$$\begin{aligned} \langle {\mathcal {A}} (u)-{\mathcal {A}} (v),u-v\rangle \ge 0; \end{aligned}$$ -
hemicontinuous if the real function \(\lambda \mapsto \langle {\mathcal {A}}(u+\lambda \, v),v\rangle \) is continuous, for every \(u,v\in V\).
Theorem A.2
Let V be a separable, reflexive Banach space and let \({\mathcal {V}} = L^p(I;V)\), for \(1<p<\infty \), where \(I=[t_0,t_1]\). Suppose that H is a Hilbert space such that V is dense and continuously embedded in H and that H is embedded into \(V^*\) according to the relation (A.2). Assume that the family of operators \({\mathcal {A}}(t,\cdot ):V\rightarrow V^*\), \(t\in I\) satisfies:
- (i):
-
for every \(v\in V\), the function \({\mathcal {A}}(\cdot ,v):I\rightarrow V^*\) is measurable;
- (ii):
-
for almost every \(t\in I\), the operator \({\mathcal {A}}(t,\cdot ):V\rightarrow V^*\) is monotone, hemicontinuous and bounded by
$$\begin{aligned} \Vert {\mathcal {A}}(t,v)\Vert _{V^*}\le C\,\Big (\Vert v\Vert ^{p-1}_{V}+k(t)\Big ), \qquad \text{ for } v\in V\quad \text{ and } \quad k\in L^{p'}(I), \end{aligned}$$ - (iii):
-
there exist a real number \(\beta >0\) and a function \(\ell \in L^1(I)\) such that
$$\begin{aligned} \langle {\mathcal {A}}(t,v),v\rangle + \ell (t)\ge \beta \,\Vert v\Vert ^p_V, \qquad \text{ for } \text{ a. } \text{ e. } t\in I \text{ and } v\in V. \end{aligned}$$
Then for each \(f\in {\mathcal {V}}^*=L^{p'}(I;V^*)\) and \(u_0\in H\), there exists a unique \(u\in W_p(I)\) satisfying
This means that \(u\in {\mathcal {V}}\), \(u'\in {\mathcal {V}}^*\) and
Proof
The existence of a unique solution \(u\in {\mathcal {V}}\) is contained in [28, Proposition 4.1, Chapter III]. The condition (iii) is slightly different here, due to the presence of the function \(\ell (t)\), but the proof of [28, Proposition 4.1, Chapter III] goes through with minor changes. \(\square \)
In order to prove existence for our problem (A.1), we will use Theorem A.2 with the choice \(V=X_0^{s,p}(\Omega ,\Omega ')\). This is the content of the next result, which generalizes [25, Theorem 2.5]. The latter only deals with the case \(f\equiv g\equiv 0\).
Theorem A.3
Let \(p\ge 2\), let \(I = [t_0,t_1]\) and suppose that g satisfies
Suppose also that
Then for any initial datum \(u_0\in L^2(\Omega )\), there exists a unique weak solution u to problem (A.1).
Proof
We denote by \(\mathbf{g}\) the mapping \(\mathbf{g}:I\rightarrow W^{s,p}(\Omega ')\), given by \((\mathbf{{ g}}(t))(x) = g(x,t)\). For almost every \(t\in I\), we define the operator
by
It is easy to check that \({\mathcal {A}}_t(v)\in (W^{s,p}(\Omega '))^*\) whenever \(v\in X_{\mathbf{g}(t)}^{s,p}(\Omega ,\Omega ')\). Additionally, \({\mathcal {A}}_t\) is a monotone operator, see [21, Lemma 3]. We now define \({\mathcal {A}}:X_0^{s,p}(\Omega ,\Omega ')\times I\rightarrow (W^{s,p}(\Omega '))^*\) to be the operator defined by
Observe that this is well-defined, since
We next show that the operator \({\mathcal {A}}\), together with the spaces
fits into the framework of Theorem A.2. Since \(p\ge 2\) and \(\Omega '\) is bounded, \(X_0^ {s,p}(\Omega ,\Omega ')\) is dense and continuously embedded in \(L^2(\Omega )\). This follows from Hölder’s inequality and the fact that smooth compactly supported functions are dense in both spaces. Note that \({\mathcal {A}}\) inherits the property of monotonicity from \({\mathcal {A}}_t\) since
We next claim that
We have
The first term on the right-hand side of (A.6) can be bounded by
using Hölder’s inequality. For the second term we observe that, when \(x\in \Omega \) and \(y\in {\mathbb {R}}^N{\setminus } \Omega '\),
where \(C>1\) depends only on the distance between \(\Omega \) and \(\Omega '\). Since \(1/(1+|y|^{N+s\,p})\in L^1({\mathbb {R}}^N)\), the second term in the right-hand side of (A.6) can be estimated by
where we used the continuous inclusion \(W^{s,p}(\Omega ')\subset L^p(\Omega ) \). This finally shows (A.5). Observe that
thanks to the assumptions on g. Thus in order to verify (ii) of Theorem A.2, we are left with proving hemicontinuity. For this, fixed \(t\in I\) and \(\lambda ,\lambda _0\in {\mathbb {R}}\), we consider
In order to show that this differences goes to 0 as \(\lambda \) goes to \(\lambda _0\), it is sufficient to write
and then use [21, Lemma 3]. This proves that \({\mathcal {A}}\) is hemicontinuous for almost every \(t\in I\).
Finally, as for hypothesis (iii) of Theorem A.2, we observe that if \(v\in X_0^{s,p}(\Omega ,\Omega ')\), then by using Poincaré inequality we have
for a constant \(C=C(N,p,s,\Omega ,\Omega ')>0\). Additionally, using Hölder’s inequality and Young’s inequality, we obtain
By combining this with the previous estimate, hypothesis (iii) of Theorem A.2 is checked. According to (A.3), \(g\in C(I;L^2(\Omega ))\) and we may define \(g_0={\mathbf {g}}(t_0)\) in \(L^2(\Omega )\). From Theorem A.2, for every \(u_0\in L^2(\Omega )\) we obtain a unique solution
to the problem
Observe that again by (A.3), we also have \(v\in C(I;L^2(\Omega ))\). Since v is a solution, we have
for every \(\phi \in L^{p}(I;X_0^{s,p}(\Omega ,\Omega '))\). Upon setting \(u=v+g\), we find that
and it verifies
for every \(\phi \in L^{p}(I;X_0^{s,p}(\Omega ,\Omega '))\). In particular, if we take \(J=[T_0,T_1]\subset I\) and \(\phi \in L^p(J;X_0^{s,p}(\Omega ,\Omega '))\), by extending \(\phi \) to be 0 outside J we get
If now the test function \(\phi \) is further supposed to belong to \(L^{p}(J;X_0^{s,p}(\Omega ,\Omega '))\cap C^1(J;L^2(\Omega ))\), by using (A.4) we can integrate by parts
Thus we obtained
for every \(J=[T_0,T_1]\subset I\) and every \(\phi \in L^{p}(J;X_0^{s,p}(\Omega ,\Omega '))\cap C^1(J;L^2(\Omega ))\). By recalling the definition of \({\mathcal {A}}_t\), this shows u is a weak solution of (A.1). \(\square \)
Proposition A.4
(Comparison principle) Let \(p\ge 2\), let \(I = [t_0,t_1]\) and suppose that g satisfies
Given an initial datum \(u_0\in L^2(\Omega )\), we consider the unique weak solution u to the initial boundary value problem
If there exists \(M\in {\mathbb {R}}\) such that
then we also have
Proof
We take \(J=[T_0,T_1]\Subset (t_0,t_1)\), by proceeding as in the first part of Lemma 3.3, we obtain
for every \(\phi \in L^{p}((-1,0);X_0^{s,p}(\Omega ,\Omega '))\cap C^1((-1,0);L^2(\Omega ))\). We still use the notation \(\phi ^\varepsilon \) and \(u^\varepsilon \) for the convolution in the time variable, as defined in (3.3). Moreover, we still indicate by \(\Sigma (\varepsilon )\) the error term (3.7). We now take the test functionFootnote 8
Observe that this function is only Lipschitz in time, but it is not difficult to see that Lipschitz functions are still feasible test functions (by a simple density argument). This gives
On the other hand
By taking the limit as \(\varepsilon \) goes to 0, we thus get
By using that (see [5, Lemma A.2])
we thus get
This is valid for every \(t_0<T_0<T_1<t_1\). By using the Monotone Convergence Theorem on the left-hand side and the fact that \(u\in C(I;L^2(\Omega ))\) on the right-hand side, we can pass to the limit as \(T_0\) goes to \(t_0\) and obtain
We used that \(u_0\le M\) on \(\Omega \), by assumption. This implies that
Since \(T_1\) is arbitrary, we finally get that
This concludes the proof. \(\square \)
As a straightforward consequence of the previous result, we get the following
Corollary A.5
(Global \(L^\infty \) estimate) Under the assumptions of Proposition A.4, assume further that
Then
Proof
By using Proposition A.4 with
we get \(u\le M\). To get the lower bound, it is sufficient to observe that \(-u\) solves the initial boundary value problem for the same equation, with data \(-g\le M\) and \(-u_0\le M\). By Proposition A.4 again, we get \(-u\le M\), as well. \(\square \)
We also include the following comparison principle with bounded subsolutions.
Proposition A.6
(Comparison with subsolutions) Let \(p\ge 2\), \(I = [t_0,t_1]\) and suppose that \(v\in L^\infty (I;L^\infty ({\mathbb {R}}^N))\) is a local weak subsolution in \(\Omega \times I\) satisfying
Consider the unique weak solution u to the initial boundary value problem
Then
Proof
The proof is almost identical with the proof of Proposition A.4. We give some details below. Take \(J=[T_0,T_1]\Subset (t_0,t_1)\). Again, as in the first part of Lemma 3.3, we obtain
for every non-negative \(\phi \in L^{p}((-1,0);X_0^{s,p}(\Omega ,\Omega '))\cap C^1((-1,0);L^2(\Omega ))\). The quantity \(\Sigma (\varepsilon )\) is still defined in (3.7), with \(v-u\) in place of u. Observe that now we have an inequality, since v is merely a subsolution. Take the test function
This gives
As before, the terms
go to zero, as \(\varepsilon \) goes to 0. Therefore, by taking the limit as \(\varepsilon \) goes to 0, we arrive at
By [4, Lemma A.3], we have
for some \(C=C(p)>0\). Then
for every \(t_0<T_0<T_1<t_1\). We can now let \(T_0\) converge to \(t_0\) and obtain
This implies
Since \(T_1\) is arbitrary, this entails the desired result. \(\square \)
Appendix B. Some complements to the proof of Lemma 3.3
We keep on using the same notation of Lemma 3.3. For every \(0<|h|<h_0/4\) and \(0<\varepsilon <\varepsilon _0\), we set
and
Then
We need to show that
We start by splitting the integral as follows
We used that \(\eta \) vanishes on \({\mathbb {R}}^N{\setminus } B_{2-2\,h}\). We now observe that
where we used the properties of convolutions, the fact that F is locally Lipschitz and the uniform \(L^\infty \) bound (3.12). Thus, up to extracting a subsequence, we can infer weak convergence in
of
to the function
By definition, this is the same as saying that the function
weakly converges in \(L^p([T_0,T_1];L^p(B_{2-2\,h}\times B_{2-2\,h}))\). This permits to conclude that
thanks to the fact that
belongs to \(L^{p'}([T_0,T_1];L^{p'}(B_{2-2\,h}\times B_{2-2\,h}))\).
For \(\Theta _2(\varepsilon )\) we use a similar argument. More precisely, we observe that if we set
we have
By using the definition of local weak solution, this implies that \({\mathfrak {F}}\in L^1([T_0,T_1]\times B_{2-2\,h})\). On the other hand, for \(t\in [T_0,T_1]\) and \(x\in B_{2-2\,h}\) we have
By recalling (3.12), this implies that
is uniformly bounded in \(L^\infty ([T_0,T_1]\times B_{2-2\,h})\). The last two facts implies that
up to extracting a subsequence. In the exact same way, we can show that
This in turn permits to infer that \(\Theta _2(\varepsilon )\) goes to 0, as well. This concludes the proof of Lemma 3.3.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Brasco, L., Lindgren, E. & Strömqvist, M. Continuity of solutions to a nonlinear fractional diffusion equation. J. Evol. Equ. 21, 4319–4381 (2021). https://doi.org/10.1007/s00028-021-00721-2
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-021-00721-2