Abstract
In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities \(\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,\) involving time-fractional Caputo derivative \(\mathcal {D}_{0|t}^{\alpha }\) and space-fractional p-Laplacian operator \((-\varDelta )^s_p\). We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of \(\gamma ,\mu ,m\) and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.
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1 Introduction
In this paper, we study the initial-boundary value problem for the nonlinear time-space fractional diffusion equation
where \(\varOmega \subset \mathbb {R}^N\) is a smoothly bounded domain; \(s\in (0,1), p\ge 2, m>0, q\ge 1\), \(\gamma ,\mu \in \mathbb {R}\) and \(\mathcal {D}_{0|t}^{\alpha }\) is the left Caputo fractional derivative of order \(\alpha \in (0,1)\) (see Definition 3).
In recent years, the study of differential equations using non-local fractional operators has attracted a lot of interest. The time-space fractional diffusion equations could be applied to a wide range of applications, including finance, semiconductor research, biology and hydrogeology, continuum mechanics, phase transition phenomena, population dynamics, image process, game theory and Lévy processes, (see [3, 5, 9, 15, 17, 22, 23]) and the references therein. When a particle flow spreads at a rate that defies Brownian motion theories, both time and spatial fractional derivatives (see [16, 24, 35]) can be employed to simulate anomalous diffusion or dispersion. Recently, motivated by some situations arising in the game theory, nonlinear generalizations of the fractional Laplacian have been introduced, (see [6, 9]).
Later on, the fractional version of the p-Laplacian was studied through energy and test function methods by Chambolle and al. in [10]. The viscosity version of this non-local operator was given by Ishii and al. in [20], Bjorland and al. in [6].
In the case \(\alpha =s=1,\,\gamma =-1\) the problem (1.1) coincides with a quasilinear parabolic equation which has been studied by Li et al in [25]. By using a Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blow-up and extinction have been classified completely in the different ranges of reaction exponents.
Moreover, when \(\alpha =s=1, m>1\) and the coefficients are \(\gamma >0, \mu =0\), the problem (1.1) was considered by Yin and Jin in [38]. They determined the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary value problem.
Vergara and Zacher in [36] have considered nonlocal in time semilinear subdiffusion equations on a bounded domain,
where the coefficients \(A=(a_{ij})\) were assumed to satisfy
They proved a well-posedness result in the setting of bounded weak solutions and studied the stability and instability of the zero function in the special case where the nonlinearity vanishes at 0. In addition, they established a blow-up result for positive convex and superlinear nonlinearities.
Later on, Alsaedi et al. [2] have studied the KPP-Fisher-type reaction-diffusion equation, which is the problem (1.1) in the case \(p=2, \gamma =-1, q=3\) and \(\mu =m=1\), in a bounded domain. Under some conditions on the initial data, they have showed that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions was analysed.
Recently, in [34], Tuan, Au and Xu studied the initial-boundary value problem for the fractional pseudo-parabolic equation with fractional Laplacian
where \(s\in (0,1), m>0\) is a constant, and \(\mathcal {N}(u)\) is the source term satisfying one of the following conditions:
-
(a)
\(\mathcal {N}(u)\) is a globally Lipschitz function;
-
(b)
\(\mathcal {N}(u)=|u|^{p-2}u,\,p\ge 2;\)
-
(c)
\(\mathcal {N}(u)=|u|^{p-2}u\log |u|,\,\,p\ge 2.\)
For the above cases, they proved the existence of a unique local mild solution and finite time blow-up solution to equation (1.3). Because of the nonlocality of the equation, the authors believe that proving the existence of a weak solution using the Galerkin method for equation (1.3) is problematic.
Motivated by the above results, in this paper we consider the time and space fractional quasilinear parabolic equation (1.1).
Using the Galerkin method, we prove the existence of a local weak solution to problem (1.1).
This, in turn, partially answers the question posed in [34] about the existence of a local weak solution to the fractional pseudo-parabolic equation. In addition, a comparison principle to problem (1.1) is obtained, and we have investigated results on the blow-up and global solution using this concept.
2 Preliminaries
2.1 The fractional Sobolev space
In this subsection, let us recall some necessary definitions and useful properties of the fractional Sobolev space.
Let \(s\in (0,1)\) and \(p\in [1,+\infty )\) be real numbers, and let the fractional critical exponent be defined as \(\displaystyle p_c^*=\frac{Np}{N-sp}\) if \(sp<N\) or \(p_c^*=\infty \), otherwise.
One defines the fractional Sobolev space as follows
This is the Banach space between \(L^p(\mathbb {R}^N)\) and \(W^{1,p}(\mathbb {R}^N)\), endowed with the norm
Let \(\varOmega \) be an open set in \(\mathbb {R}^N\) and let \(\mathcal {W}=(\mathbb {R}^N\times \mathbb {R}^N)\backslash ((\mathbb {R}^N\backslash \varOmega )\times (\mathbb {R}^N\backslash \varOmega ))\). It is obvious that \(\varOmega \times \varOmega \) is strictly contained in \(\mathcal {W}\).
Denote
The space \(W^{s,p}(\varOmega )\) is also endowed with the norm
where the term
is the so-called Gagliardo semi-norm of u, which was introduced by Gagliardo [13] to describe the trace spaces of Sobolev maps.
We refer to [28] and [7], where one can find a description of the most useful properties of the fractional Sobolev spaces \(W^{s,p}(\varOmega )\). In the literature, fractional Sobolev-type spaces are also called Aronszajn, Gagliardo or Slobodeckij spaces, by the name of the people who first introduced them, practically concurrently (see [4, 14, 31]).
For Gagliardo semi-norms, the next result is a Poincaré inequality. This is standard, but we should also always pay careful attention to the sharp constants dependence on s.
Lemma 1
([28], Theorem 6.7) Let \(s\in (0,1)\) and \(p\in [1,+\infty )\) be such that \(sp<N\). Let \(\varOmega \subseteq \mathbb {R}^N\) be an extension domain for \(W^{s,p}\). Then, there exists a positive constant \(C=C(N,p,s,\varOmega )\) such that, for any \(u\in W^{s,p}(\varOmega )\), we have
for any \(q\in [p,p^*],\) where \(\displaystyle p^*=p^*(N,s)=\frac{Np}{N-sp}\) is the so-called fractional critical exponent.
That means, the space \(W^{s,p}(\varOmega )\) is continuously embedded in \(L^q(\varOmega )\) for \(q\in [p,p^*].\) If, in addition, \(\varOmega \) is bounded, then, the space \(W^{s,p}(\varOmega )\) is continuously embedded in \(L^q(\varOmega )\) for \(q\in [1,p^*].\)
Lemma 2
([29], Lemma 2.1. Fractional Gagliardo–Nirenberg inequality) Let \(p>1, \tau >0, N,q\ge 1, 0<s<1\) and \(0 <a\le 1\) be such that
We have
for some positive constant C independent of u.
2.2 Fractional operators
This part is devoted to the definitions and properties of fractional derivatives in time and space.
Definition 1
([21], p. 69) The left and right Riemann-Liouville fractional integrals of order \(0<\alpha <1\) for an integrable function u(t) are given by
and
Definition 2
([21], p. 70) The left and right Riemann-Liouville fractional derivatives \(\mathbb {D}_{0|t}^{\alpha }\) of order \(\alpha \in (0,1)\), for an absolutely continuous function u(t) is defined by
and
Lemma 3
([21], Lemma 2.20) If \(\alpha >0,\) then for \(u\in L^1(0,T)\), the relations
are true.
Definition 3
([21], p. 91) The \(\alpha \in (0,1)\) order of left and right Caputo fractional derivatives for \(u\in C^1([0,T])\) are defined, respectively, by
and
If \(u\in C^1([0,T])\), then the Caputo fractional derivative can be represented by the Riemann-Liouville fractional derivative in the following form
and
Lemma 4
([39], Corollary 4.1) Let \(T>0\) and let U be an open subset of \(\mathbb {R}\). Let further \(u_0\in U,\) \(k\in H^1_1(0,T), H\in C^1(U)\) and \(u\in L^1(0,T)\) with \(u(t)\in U,\) for a. a. \(t\in (0,T)\). Suppose that the functions \(H(u), H'(u)u\), and \(H'(u)(k_t*u)\) belong to \(L^1(0,T)\) (which is the case if, e.g., \(u\in L^\infty (0,T)\)). Assume in addition that k is nonnegative and nonincreasing and that H is convex. Then
Lemma 5
([1], Lemma 1) For \(0<\alpha <1\) and any function u(t) absolutely continuous and real-valued on [0, T], one has the inequality
Property 1
([21], p. 95-96) If \(0<\alpha <1\), \(u\in AC^1[0,T]\) or \(u\in C^1[0,T]\), then
and
hold almost everywhere on [0, T]. In addition,
Property 2
([21], Lemma 2.7) Let \(0<\alpha <1\) and \(u\in C^1[0,T], \varphi \in L^p(0,T)\). Then the integration by parts for Caputo fractional derivatives has the form
Definition 4
([32]. Lemma 5.1) The fractional p-Laplacian operator for \(s\in (0,1), p>1\) and \(u\in W^{s,p}(\varOmega )\), is defined by
where
is a normalization constant and “P.V.” is an abbreviation for “in the principal value sense”. Since they will not play a role in this work, we omit the P.V. sense. However, let us stress that these constants guarantee:
Definition 5
([26], Theorem 5) We say that \(u\in W^{s,p}_0(\varOmega )\) is an (s, p) - eigenfunction associated to the eigenvalue \(\lambda \) if u satisfies the Dirichlet problem
weakly, it means that
for every \(\psi \in W^{s,p}_0(\varOmega )\). If we set as
then the nonlinear Rayleigh quotient determines the first eigenvalue
Lemma 6
([26], Lemma 15) Assume that for all j, if we have
Then
Note that the minimization problem is not quite the same if \(\mathbb {R}^N\times \mathbb {R}^N\) is replaced by \(\varOmega \times \varOmega \) in the integral. This choice has the advantage that the property
is evident for subdomains. By changing coordinates it implies
This asserts that small domains have large first eigenvalues (see [26] references therein).
Lemma 7
([7], Lemma 2.4. Fractional Poincaré inequality)
Let \(1 \le p <\infty \) and \(s \in (0, 1)\), \(\varOmega \subset \mathbb {R}^N\) be an open and bounded set. Then, it holds
and we have the lower bound
where the geometric quantity \(\mathcal {I}_{N,s,p(\varOmega )}\) is defined by
We define the inner product of the operator \((-\varDelta )^s_p\) for \(u,v\in W^{s,p}(\varOmega )\) as
Definition 6
A function \(u=u(x,t)\in W^{s,p}(\varOmega ;L^\infty (0,T))\cap L^2(\varOmega ;L^\infty (0,T))\) is called a weak solution of (1.1) if the following identity holds
almost everywhere in \(t\in [0,T]\), for any \(\varphi =\varphi (x,t)\in W_0^{s,p}(\varOmega ;L^\infty (0,T))\), such that \(\varphi \ge 0\) in \(\varOmega \), \(\varphi =0\) on \(\partial \varOmega \).
2.3 Notations
We recall standard notations, which will be used in the sequel. If \(\varOmega \) is a bounded and open set in \(\mathbb {R}^N\) \((\varOmega \subseteq \mathbb {R}^N)\), we denote
We include the following function space
with the norm
3 A comparison principle
In this section we study a comparison principle for the fractional parabolic equation. We begin by presenting a weak subsolution and a weak supersolution to the problem (1.1).
Definition 7
A real-valued function
is called a weak subsolution of (1.1) if the inequality
holds for any \(\varphi \in W_0^{s,p}(\varOmega ;L^\infty (0,T))\), such that \(\varphi \ge 0\) in \(\varOmega \), \(\varphi =0\) on \(\partial \varOmega \).
Similarly, a real-valued function
is called a weak supersolution of (1.1) if it satisfies the inequality
A function is a weak solution, if it is both a weak subsolution and a weak supersolution.
Theorem 1
Let \(s\in (0,1), p\ge 2\) and let \(m, q, \gamma , \mu \) satisfy one of the following conditions:
Suppose that \(u, v\in \varPi \) be real-valued weak subsolution and weak supersolution of (1.1), respectively, with \(u_0(x)\le v_0(x)\) for \(x\in \varOmega \). Then \(u\le v\) a.e. in \(\varOmega _T\).
Corollary 1
Assume that \(p\ge 2\) and let \(m, q, \gamma , \mu \) satisfy the conditions in Theorem 1. If \(u_0(x)\ge 0\) for all \(x\in \varOmega ,\) then \(u(x,t)\ge 0,\,x\in \varOmega ,\,t\ge 0.\)
The proof of Corollary 1 follows from Theorem 1. More precisely, if \(u_0(x)\ge 0\), taking 0 as a subsolution, then we have \(u(x,t)\ge 0\).
Proof of Theorem 1
We choose the test function \(\varphi =(u-v)_+\), where \((u-v)_+\) is the positive part of a real quantity \((u-v)_+=\max \{u-v, 0\}\). Then it follows that \(\varphi (x,0)=0,\) \( \varphi (x,t)|_{\partial \varOmega }=0\). By subtracting (3.2) from (3.1), we obtain for \(t\in (0,T]\)
According to (2.3), we can write the last term of the left-hand side inequality (3.3) in the form
where
Hence, we can show that
is nonnegative for any \(p\ge 2\), thanks to the inequality (see [27], P. 99)
with \(a:=u(x)-u(y),\, b:=v(x)-v(y)\) in (3.5).
Now, we will evaluate the right-side of (3.3).
Taking account the following inequality
where \(L(m)= C(m)\max (\Vert u\Vert ^{m-1}_{L^\infty (\varOmega )},\Vert v\Vert ^{m-1}_{L^\infty (\varOmega )})\), we can verify that
where we have used the well known inequality [28, Theorem 8.2] for any \(u\in L^p(\varOmega )\) such that
which gives the boundness of \(\max (\Vert u\Vert ^{m-1}_{C(\varOmega )},\Vert v\Vert ^{m-1}_{C(\varOmega )})\).
In addition, from the Lipchitsz condition it follows that
where \(L(q)= C(q)\max (\Vert u\Vert ^{q-2}_{L^\infty (\varOmega )},\Vert v\Vert ^{q-2}_{L^\infty (\varOmega )})\). Hence, from (3.9), it follows that
Combining (3.4), (3.8) and (3.11), we can rewrite the inequality (3.3) as
Using Lemma 5, the inequality (3.12) can be rewritten in the following form
At this stage, we have to consider three cases depending on \(\gamma , \mu \):
\(\bullet \) The case \(\gamma \ge 0,\mu \ge 0.\) Applying the left Caputo fractional differentiation operator \(\mathcal {D}_{0|t}^{1-\alpha }\) to both sides of (3.13) and using Property 1, we obtain
Then, from the weakly singular Gronwall’s inequality (see [19], Lemma 7.1.1 and [18], Lemma 6, p. 33)
Finally, it follows that \(u\le v\) almost everywhere for \((x,t)\in \varOmega _T\).
\(\bullet \) The case \(\gamma \le 0,\mu \le 0.\) According to the inequality (3.13), the right-hand side integral is positive and the coefficients \(\gamma ,\mu \) are non-positive, we deduce that
Therefore, repeating the similar procedure as above we obtain
Consequently, we have \(u\le v\) almost everywhere for \((x,t)\in \varOmega _T\).
\(\bullet \) The case \(\gamma \ge 0,\mu \le 0\) or \(\gamma \le 0,\mu \ge 0\). Using the inequality (3.13), it follows that
or
respectively. By the weakly singular Gronwall’s inequality, we arrive at \(u\le v\) almost everywhere for \((x,t)\in \varOmega _T\). \(\square \)
4 Local well-posedness
4.1 Existence of a local weak solution
In this subsection, we will prove that problem (1.1) has the local weak solution by Galerkin method.
Theorem 2
Let \(u_{0} \in W^{s,p}_0(\varOmega ), u_0\ge 0,\, sp<N\) and let either \(1<m<q-1<p-1\) or \(1<q-1<m <p-1.\) Then there exists \(T>0\) such that the problem (1.1) has a local real-valued weak solution \(u\in \varPi \), where \(\varPi \) is defined in (2.4).
Proof
\(\bullet \) The case \(1<m<q-1<p-1\). The space \(W^{s,p}_0(\varOmega )\) is separable. Then there exists a countable linear set \(\{\omega _j\}_{j\in N}\) that is everywhere dense in \(W^{s,p}_0(\varOmega )\).
Let us consider the Galerkin approximations
where the unknown \(v_{nj}\in C^1([0,T_n])\) functions satisfy the following system of ordinary fractional differential equations:
supplemented by the initial condition
where
First of all, we need to prove that the system of Galerkin equations (4.2) has a solution \(v_{nj}\in C^1([0, T_n]),\,j=\overline{1,n}\) for some \(T_n > 0\), which depends on \(n\in N\). Therefore, we note that the system of equations (4.2) can be represented in the following form
where \(a_{jk}\) is an invertible matrix for each \(n\in N\) and the functions \(F_{ik}(v_n),\,i=1,2,3\) are defined by
and
Next, we will prove the functions \(F_{ik}(v_n),\,i=1,2,3\) are locally Lipschitz function. Indeed, we have
Using the following inequality, for \(p\ge 1,\)
and the generalized Hölder inequality with parameters
in the last equality, we obtain
where \(\varPhi ^{p-2}=\max \{[u_n^1]_{W^{s,p}(\varOmega )}; [u_n^2]_{W^{s,p}(\varOmega )}\}\) and \([\,\cdot \,]_{W^{s,p}(\varOmega )}\) is the Gagliardo semi-norm. Consequently,
At this stage using
in the last term of the previous inequality, and recalling \(v^1_{nj}, v^2_{nj}\in C^1([0, T_n])\) we arrive at
Accordingly, using the inequalities (3.7) and (3.9) to \(F_{2k}(v_n)\), for \(k,j=\overline{1,n}\), we deduce that
Similarly, from (3.9) and (3.10) we obtain an estimate for \(F_{3k}(v_n)\), for \(k,j=\overline{1,n}\), in the following form
From Lemma 1, the space \(W^{s,p}(\varOmega )\) is continuously embedded in \(L^2(\varOmega )\). Indeed, the right-hand side of \(F_{ik}(v_n),\,\,i=1,2,3,\,k=\overline{1,n}\) is continuous with respect to \(t\in [0, T_n]\) and locally Lipschitz function with respect to \(v_n(t)\).
Therefore, due to [21, Theorem 3.25] the Cauchy problem for the system of equations (4.4) has a unique solution \(v_{nj}\in C^1([0, T_n]),\,j=\overline{1,n}\) for some \(T_n > 0\), which depends on \(n\in N\).
Multiplying the expression (4.2) by \(v_{nk}(t)\) and performing the summation over \(k=1,...,n,\) it follows that
Applying the fractional Poincaré inequality from Lemma 7 and the inequality in Lemma 5 to the previous identity, we get
At this stage we have to consider different cases of coefficients \(\gamma \) and \(\mu \).
\(\bullet \) The case \(\gamma ,\,\mu >0\). Thanks to the inequality (see [30], P. 417),
for \(a=q-1,\,b=m,\) and \(c=2\) in (4.6) we obtain
Consequently, it follows that
Due to the inequality (4.7) for \(a=p-1,\,b=q-1\) and \(c=2\), it holds
Therefore, using the last inequality in (4.9) we get
Finally, choosing the constants \(\varepsilon , {\tilde{\varepsilon }}>0\) such that
then we get the following result
where
Define \(\displaystyle \varPhi (t):=\int _\varOmega |u_{n}|^2dx\), then applying the left Riemann-Liouville fractional integral operator \(I_{0|t}^{\alpha }\) to both sides of (4.10) and using Property 1, we get
Furthermore, according to Gronwall-type inequality for fractional integral equations (see [12], Lemma 4.3) we obtain
where \(E_{\alpha ,1 }(z)\) is the Mittag-Leffler function, defined by
Finally, in view of Corollary 1 for real-valued u, we conclude that there exists finite \(T_0>0\),
for all \(t\in [0,T]\), \(T<T_0\), where A(T) is a constant independent of n.
\(\bullet \) The case \(\gamma >0\) and \(\mu \le 0\). Then from (4.6) we obtain
Setting \(a=p-1,\,b=m,\) and \(c=2\) in (4.8) we can rewrite the last estimate as
By choosing the constants \(\varepsilon , {\tilde{\varepsilon }}>0\) which satisfy
then we get
The conclusion can be derived as in the previous case.
\(\bullet \) The case \(\gamma \le 0\) and \(\mu >0\). Accordingly from (4.6) we have
Next, choosing \(a=p-1,\,b=q-1,\) and \(c=2\) in (4.8) it follows
Now, taking \(\varepsilon , {\tilde{\varepsilon }}>0\), which satisfy
we obtain the estimate
Similarly, the conclusion can be derived as in the previous case.
\(\bullet \) The case \(\gamma ,\mu \le 0\). Take into consideration the inequality (4.6) it yields
Using the fact that \(\lambda _1\) is nonnegative we obtain
Hence, applying the left Riemann-Liouville integral \(I_{0|t}^{\alpha }\) to the last inequality and using Property 1, we deduce that
Finally, it follows that
Next, multiplying the expression (4.2) by \(\mathcal {D}_{0|t}^{\alpha }v_{nk}(t)\) and summing over \(k=\overline{1,n}\), we obtain
with
Due to Lemma 5 it follows that
Moreover the identity (4.13) becomes
At this stage, we consider the function
which is convex. By differentiating respect to \(\omega \) we have \(H'(\omega )(t)=|\omega (t)|^\frac{p-2}{2}\). From Lemma 4 for the function \(H(\omega )(t)\) we obtain the following inequality
Denote \(\omega (t)=|u_{n}(x,t)-u_{n}(y,t)|^2\). Then, we obtain
Therefore, using (4.14) and the last inequality we get
Since the operator \(\mathcal {D}_{0|t}^{\alpha }\) is with respect to the variable t it follows that
Finally, the identity (4.12) can be rewritten as
At this stage, we should study the different cases of the coefficients \(\gamma \) and \(\mu \).
\(\bullet \) The case \(\gamma ,\mu >0\). Using the Hölder and \(\varepsilon \)-Young inequalities
where \(\displaystyle C(\varepsilon )=\frac{1}{p'\varepsilon ^{p'-1}}\) for the right hand side of (4.15), respectively, we get
and
From Lemma 2 we obtain
and
Hence, from the last inequalities (4.15) we obtain
After choosing the constants \(\varepsilon , \varepsilon _1\) such that \(\displaystyle 1>\frac{1}{\varepsilon }+\frac{1}{\varepsilon _1}\), and from the estimate (4.11) it follows that
where \(\mathcal {C_*}:=\mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)+\mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)\) and \( B(T):=A({\tilde{\varepsilon }},T)+A({\tilde{\varepsilon }}_1,T).\) Therefore,
Define \(y(t):=[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\) and using the left Riemann-Liouville integral \(I_{0|t}^\alpha \) to (4.21), according to Property 1, we arrive at
which satisfies (see [33], Lemma 3.1)
Finally, we have
From the inequalities (4.20) and (4.22), we obtain
Integrating both sides of (4.23) by the left Riemann-Liouville integral \(I_{0|t}^\alpha \) and using Property 1, the last inequality becomes
Consequently, applying the left Caputo derivative \(\mathcal {D}_{0|t}^\alpha \) due to Property 1, also noting the facts that \([u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\) is bounded and \(\mathcal {D}_{0|t}^\alpha [u_{n}(\cdot , 0)]_{W^{s,p}(\varOmega )}=0\), we can establish
where L(p, T) does not dependent to n.
\(\bullet \) The case \(\gamma >0\) and \(\mu \le 0\). Accordingly, the inequality (4.15) becomes
From the estimates (4.16) and (4.18) we can rewrite the last inequality in the form
By choosing \(\varepsilon \) small enough such that \(\displaystyle \frac{1}{2}-C(\varepsilon )>0\), and using (4.11) it follows that
The conclusion can be obtained, as in the previous case.
\(\bullet \) The case \(\gamma \le 0\) and \(\mu >0\). The inequality (4.15) becomes
Using the estimates (4.17) and (4.19) we have
Taking the constant \(\varepsilon _1\) small enough such that \(\displaystyle \frac{1}{2}-C(\varepsilon _1)>0\), and noting (4.11) it follows that
The conclusion of this case also can be obtained, as in the first case.
\(\bullet \) The case \(\gamma ,\mu \le 0\). Then, the estimate (4.15) can rewritten as
Applying the left Riemann-Liouville integral \(I^\alpha _{0|t}\) to the last inequality from Property (1) it follows that
From the estimate (4.22) we arrive at
Next, using the left Riemann-Liouville fractional derivative for the last inequality, and from Property 3 and the identity
it follows that
Passing to the limit where \(n\rightarrow \infty \), from the estimates in the previous estimates, we conclude that
Consequently, from (4.26) there exists a subsequence \(\{u_{n_k}\}\) of \(\{u_n\}_{n\in \mathbb {N}}\) weak star converging to some element from \(W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T))\) such as
Similarly, from (4.27), we deduce that one can extract a subsequence \(\{u_{n_k}\}\) of \(\{u_n\}_{n\in \mathbb {N}}\) such that
Since, \(W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T))\subset L^2(\varOmega ;L^\infty (0,T))\), from (4.26) it follows that the sequences \(\{u_n\}_{n\in \mathbb {N}}\) and \(\mathcal {D}_{0|t}^{\alpha }u_{n}\) are bounded in \(L^2(\varOmega ;L^\infty (0,T))\). Then, it particular \(\{u_n\}_{n\in \mathbb {N}}\) is bounded in \(W^{s,p}(\varOmega )\). It is known by Lemma 1, that the embedding of \(W^{s,p}(\varOmega )\) in \(L^2(\varOmega )\) is continuous. It gives us that the subsequence \(\{u_{n_k}\}\) can be chosen such that \(u_{n_k}\rightarrow u\) in the norm of \(L^2(\varOmega )\), converging almost everywhere. The previous argument leads us to the limit in (4.2). However, we multiply (4.2) by \(\theta _k(t)\in C[0,T]\), then summing up both sides over \(k=\overline{1,n}\), to get
almost everywhere in \(t\in [0,T]\), where \(\displaystyle \varPsi (x,t)=\sum _{k=1}^{n}\theta _{k} (t)\omega _{k} (x) \).
Taking into account the obtained inclusions and convergence, we pass in (4.2) to the limit as \(n\rightarrow \infty \) and obtain Definition 6 for \(\varphi =\varPsi \). Since the set of all functions \(\varPsi (x,t)\) is dense in \(\varPi \), then the limit relation holds for all \(\varphi =\varphi (x,t)\in W_0^{s,p}(\varOmega ;L^p(0,T)).\)
\(\bullet \) The case \(1<q-1<m <p.\) We repeat the entire procedure described above by simply changing the condition inequality (4.7) to \(1<q-1<m<p.\) \(\square \)
4.2 Uniqueness of a weak solution
In this subsection we discuss the uniqueness of weak solutions.
Theorem 3
Let \(u_0\in W^{s,p}_0(\varOmega ), u_0\ge 0\) and \(sp<N\). Then the local real-valued weak solution of (1.1) on (0, T), \(T<\infty \), is unique.
Proof
Assume that we have two real-valued weak solutions u and v for problem (1.1). Hence, by Definition 6, we obtain
and
By subtracting the previous two inequalities, it follows for \(t\in (0,T]\) that
Using the fact that \(\mathcal {C}\) is nonnegative from (3.6), and the estimates (3.8), (3.11) for \(\mathcal {A}, \mathcal {B}\), respectively, we deduce that
At this stage choosing the real-valued test function
and using Lemma 5, we can rewrite the last inequality as
Therefore, we should consider three cases depending on \(\gamma , \mu \). By repeating the entire procedure as in the proof of Theorem 1, we obtain the main inequality
which is equivalent to \((u-v)_+=0\). Finally, we conclude that \(u=v\). \(\square \)
5 Global existence and blow-up of solutions
5.1 Blow-up of solution
In this subsection we will show the blow-up of solution to (1.1) using the comparison principle.
Let \(\xi (x)>0\) and \(\lambda _1(\varOmega )>0\) be the first eigenfunction and the first eigenvalue [26, Theorem 5], respectively, related to the Dirichlet problem:
with \(\Vert \xi \Vert ^2_{L^2(\varOmega )}= 1.\)
Theorem 4
Let \(p\ge 2,\) \(u_0>0\), and assume that one of the following conditions holds:
(a) \(p=q\ge 2, m>1\) and \(\lambda _1(\varOmega )\ge \mu , \gamma >0\);
(b) \(p-1=m\ge 1, q>2\) and \( \lambda _1(\varOmega )\ge \gamma , \mu >0\);
(c) \(p\ge 2, m>1, q\ge 1\) and \(\lambda _1(\varOmega ),\gamma >0, \mu \le 0\);
(d) \(p\ge 2, m+1=q>2\) and \(\gamma , \mu , \lambda _1(\varOmega )>0.\)
Then the positive solution u(x, t) of (1.1) blows up in finite time
where \(k=m-1\) in cases (a), (c), (d) and \(k=q-2\) in cases (b), (d) and \(\alpha \in (0,1)\), \(\varGamma \) is the Euler Gamma function, namely, we have
Proof
First we will prove the cases (a) and (b).
We shall prove this theorem by constructing a proper weak subsolution to (1.1). We will seek the solution \(v(x,t)=\xi (x)f(t)>0\) with the initial data \(v_0(x)=\xi (x)f(0),\) such that \(0\le v_0(x)\le u_0(x)\) on \(x\in \varOmega \). Multiplying the equation (1.1) by v(x, t), and integrating the equality over \(\varOmega \), one obtains
Hence, from Lemma 5, it follows that
At this stage, by denoting \(f^2(t)=z(t)\), we have to consider the cases:
(a) If \(p=q \ge 2, m>1\) and \(\lambda _1(\varOmega ) \ge \mu , \gamma >0\), then (5.2) can rewritten as
Using the idea of paper [11], we set for any \(t\in (0,b)\),
Accordingly, we have the initial condition \(z(0)=z_0>0\). We should note that the function z(t), \(\lim _{t\rightarrow b^-}z(t)\rightarrow \infty ,\) diverges at \(t=b\). Moreover, for any \(t\in (0,b)\) and any \(\tau \in (0,t)\) we can obtain
From Definition 3 it follows for all \(t\in (0,b)\),
Therefore, z(t) diverges at \(t=b\) yielding that
(b) If \(p-1=m\ge 1, q>2\) and \( \lambda _1(\varOmega )\ge \gamma , \mu >0\), then from (5.2) we obtain
We can argue as the previous case by choosing for any \(t\in (0,b)\) the function
Similarly, for any \(t\in (0,b)\) and any \(\tau \in (0,t)\), we obtain
Finally, z(t) diverges at \(t=b\) for
(c) For \(p\ge 2, m>1, q\ge 1\) and \(\lambda _1(\varOmega ),\gamma >0, \mu <0\), inequality (5.2) yields
(d) For \(p\ge 2, m+1=q>2\) and \( \gamma , \mu , \lambda _1(\varOmega )>0\), using (5.2) we have
Proof of (c) and (d) can be derived from the previous cases. We just omit it. The proof is complete. \(\square \)
5.2 Global solution
In this subsection, we prove the existence of global solutions of problem (1.1).
Theorem 5
Assume that \(u_0\in W^{s,p}_0(\varOmega )\cap L^\infty (\varOmega ),\,s\in (0,1),\, u_0\ge 0\), and let \(p, q, m, \gamma , \mu \) satisfy one of the following conditions:
(a) \(p=m+1=q>2\) and \(0<\gamma +\mu \le \lambda _1(\varOmega );\)
(b) \(p=q\) or \(p=m+1\) and \(0\le \gamma , \mu \le \lambda _1(\varOmega );\)
(c) \(p\le m+q\) and \(\gamma ,\mu \in \mathbb {R};\)
(d) \(p\ge 2, m>1, q\ge 1\) and \(\gamma ,\mu \le 0;\)
(e) \(p=q, m>1\) and \(\gamma \le 0,\,\mu >0\).
Then the problem (1.1) admits a global in time positive solution.
Remark 1
Note that in the limiting case \(\alpha \rightarrow 1\) and \(s\rightarrow 1,\) the results of Theorem 5 coincides with the results obtained in [25].
Proof of Theorem 5
(a) Let \(\varOmega ^*\subset \mathbb {R}^N\) be a smooth domain such that \(\varOmega \subset \subset \varOmega ^*\).
Define \(\psi \) and \(\lambda _1(\varOmega ^*)\) to be the first eigenfunction and the first eigenvalue related to the Dirichlet problem:
with \(\displaystyle \int _{\varOmega ^*}|\psi (x)|^{p}dx=1\), for more details see [26, Lemma 15]. Then, from Lemma 6 we have \(\lambda _1(\varOmega ^*)\le \lambda _1(\varOmega )\), where \(\lambda _1(\varOmega )\) is the first eigenvalue of (2.2). Moreover, in view of [26, Theorem 16], we can choose a suitable \(\varOmega ^*\) and \(\theta >0\) which satisfies \(\theta \le \lambda _1(\varOmega ^*)\le \lambda _1(\varOmega )\) . Therefore, let K be so large such that
where \(\beta =\inf _{\varOmega }\psi >0,\) which we note that \(\psi >0\) in \(\varOmega \) from the results of Lindgren and Lindqvist in [26, Theorem 5]. Following that, a simple calculation shows that for each nonnegative test-function \(\varphi =\varphi (x,t)\in \varPi \cap W_0^{s,p}(\varOmega ;L^\infty (0,T))\), we have
where \(\langle \cdot ,\cdot \rangle \) is the inner product. Hence, noting that \(p=m+1=q>2,\) and choosing \(\theta :=\gamma +\mu \), the last identity takes the form
It follows that \(w=K\psi \) is a weak supersolution of problem (1.1). From Theorem 1, we have \(0\le u\le w\) almost everywhere in \(\varOmega _T\). It is also important to note that the function w is independent of t, allowing us to continue the method at any time interval \([T, T']\). As a result, we may say that the solution to (1.1) is global in time.
(b) Due to the expression (5.3) and the conditions \(p=q\) or \(p=m+1\), it follows that
and
Since, \(0\le \gamma ,\mu \le \lambda _1(\varOmega )\), then the function \(w=K\psi \) is also a weak supersolution of (1.1). The conclusion is established using the same argument as before.
(c) From Definition 5 assume that u is an eigenfunction associated to the eigenvalue \(\lambda _1(\varOmega )\), which is a nonnegative [26, Theorem 5]. Then by (5.3) it follows that
Choosing constants \(r, r'\) such as
we obtain
Using the Hölder and \(\varepsilon \)-Young inequalities to the last expression, it follows that
Therefore, the identity (5.4) becomes
Now, taking \(\varepsilon \) small enough, such that \(\lambda _1(\varOmega )\varepsilon -\gamma \ge 0\) and \(\lambda _1(\varOmega ) C(\varepsilon )-\mu \ge 0\), we can get that the last inequality will be non-positive
which completes our proof by the comparison principle.
(d) We proceed by multiplying each term of (1.1) by \(u\ge 0\) and then integrating over \(\varOmega \). Thus, we obtain
and taking into account that \(\gamma ,\mu \le 0,\) \(\langle (-\varDelta )^s_pu,u\rangle \ge 0\), it follows that
By Lemma 5, it implies
Moreover, the Caputo derivative depends on the variable t, and the last expression can be rewritten as
Hence, applying the left Riemann-Liouville integral \(I_{0|t}^{\alpha }\) to the inequality (5.6) and using Property 1, we obtain
Finally, using \(u_0\ge 0\) and Corollary 1, we get
(e) Without loss of generality, for \(\gamma \le 0,\,\mu >0\) we can get from (5.5), by (2.3), that
Then, by Lemma 7 for \(p=q\), we obtain
Using the fact that \(\lambda _1(\varOmega )\) coincides with the sharp constant in Lemma 7 [8, page 2] we choose the domain such that \(\displaystyle C_{N,s,p}\ge \mu \lambda _1(\varOmega )\ge \frac{\mu }{\mathcal {I}_{N,s,p(\varOmega )}}\) holds, which gives us
Accordingly, the conclusion follows as in the previous case. \(\square \)
5.3 Asymptotic behavior of solution
In this subsection, we give the time-decay estimates of global solutions of problem (1.1).
Theorem 6
Assume that \(u_0>0\) and that one of the following conditions holds:
(a) \(m=q-1>0\) and \(\gamma +\mu < 0;\)
(b) \(m>0, q>1\) and \(\gamma < 0,\,\mu =0;\)
(c) \(m>0, q>1\) and \(\gamma =0,\,\mu < 0\).
Then the positive global solution to problem (1.1) satisfies the estimate
where M is a positive constant dependent of \(u_0,\) and \(r=m\) in cases (a), (b) and \(r=q-1\) in cases (a), (c).
Proof
(a) Let us consider the function \(v(x,t):=v(t)>0\) for all \(x\in {\overline{\varOmega }}\). Then it follows that
According to the fact that \((-\varDelta )^s_pv(t)=0\) and \(m=q-1>0\), \(\gamma +\mu < 0\), the last expression can be rewritten in the following form
which ensures that v(t) satisfies (1.1) with the initial data \(0<\max \limits _{x\in \varOmega }u_0(x)\le v_0\).
It is known from the results of Zacher and Vergara in [37, Theorem 7.1], that if \(v_0>0, \nu>0, m>0\), then the solution to equation (5.7) satisfies estimate \(v(t)\le \frac{M}{1+t^\frac{\alpha }{r}},\) for all \(t\ge 0.\) As \(0<u_0(x)\le v_0,\) then v(t) is a supersolution of problem (1.1). This completes the proof.
Cases (b) and (c) are proved in a similar way, completely repeating the above calculations.
The proof is complete. \(\square \)
References
Alsaedi, A., Ahmad, B., Kirane, M.: A survey of useful inequalities in fractional calculus. Fract. Calc. Appl. Anal. 20(3), 574–594 (2017). https://doi.org/10.1515/fca-2017-0031
Alsaedi, A., Kirane, M., Torebek, B.T.: Global existence and blow-up for a space and time nonlocal reaction-diffusion equation. Quaest. Math. 44(6), 747–753 (2021)
de Andrade, B., Siracusa, G., Viana, A.: A nonlinear fractional diffusion equation: Well-posedness, comparison results, and blow-up. J. Math. Anal. Appl. 505(2), 125524 (2022)
Aronszajn, N.: Boundary values of functions with finite Dirichlet integral. Tech. Report of Univ. of Kansas 14, 77–94 (1955)
Bertoin J.: Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge 121 (1996)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-local gradient dependent operators. Adv. Math. 230, 1859–1894 (2012)
Brasco, L., Lindgren, E., Parini, E.: The fractional Cheeger problem. Interfaces Free Bound. 16, 419–458 (2014)
Brasco, L., Parini., E.: The second eigenvalue of the fractional \(p\)-Laplacian. Adv. Calc. Var. 9(4), 323–355 (2016)
Caffarelli, L.: Nonlocal equations, drifts and games. Nonlinear Partial Differential Equations, Abel Symposia 7, 37–52 (2012)
Chambolle, A., Lindgren, E., Monneau, R.: A Hölder infinity Laplacian. ESAIM Control Optim. Calc. Var. 18, 799–835 (2012)
Coclite, G.M., Dipierro, S., Maddalena, F., Valdinoci, E.: Singularity formation in fractional Burgers’ equations. J. Nonlinear Sci. 30, 1285–1305 (2020)
Diethelm, K., Ford, N.J.: Multi-order fractional differential equations and their numerical solution. Appl. Math. Comput. 154(3), 621–640 (2004)
Gagliardo, E.: Caratterizzazioni delle tracce sulla frontiera relative ad alcune classi di funzioni in \(n\) variabili. Rend. Sem. Mat. Univ. Padova 27, 284–305 (1957)
Gagliardo, E.: Proprietà di alcune classi di funzioni in piú variabili. Ric. Mat. 7, 102–137 (1958)
Gal, C. G., Warma, M.: Fractional-in-Time Semilinear Parabolic Equations and Applications. Springer Nature Switzerland AG (2020)
Giga, Y., Namba, T.: Well-posedness of Hamilton-Jacobi equations with Caputo’s time fractional derivative. Comm. Partial Differential Equations 42(7), 1088–1120 (2017)
Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7, 1005–1028 (2008)
Haraux, A.: Nonlinear Evolution Equations - Global Behavior of Solutions. Lecture Notes in Mathematics, vol. 841. Springer-Verlag, Berlin-New York (1981)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer-Verlag, Berlin-New York (1981)
Ishii, H., Nakamura, G.: A class of integral equations and approximation of \(p\)-Laplace equations. Calc. Var. Partial Differential Equations 37, 485–522 (2010)
Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies (2006)
Laskin, N.: Fractional quantum mechanics and Lévy path integrals. Phys. Lett. A. 268, 298–305 (2000)
Li, L., Liu, J., Wang, L.: Cauchy problems for Keller-Segel type time-space fractional diffusion equation. J. Differential Equations 265(3), 1044–1096 (2018)
Nane, E.: Fractional Cauchy Problems on Bounded Domains: Survey of Recent Results. Fractional Dynamics and Control. Springer, New York (2012)
Li, Y., Zhang, Zh., Zhu, L.: Classification of certain qualitative properties of solutions for the quasilinear parabolic equations. Sci. China Math. 61, 855–868 (2018)
Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differential Equations 49, 795–826 (2014)
Lindqvist, P.: Notes on the Stationary \(p\)-Laplace Equation. Springer (2019)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Nguyen, H.: Squassina. M.: Fractional Caffarelli-Kohn-Nirenberg inequalities. J. Funct. Anal. 274, 2661–2672 (2018)
Quittner, P., Souplet, P.: Superlinear Parabolic Problems, Blow-Up, Global Existence and Steady States, Second ed., Birkhäuser (2019)
Slobodeckij, L.N.: Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations. Leningrad. Gos. Ped. Inst. Učep. Zap. 197, 54–112 (1958)
del Teso, F., Gómez-Castro, D., Vázquez, J.L.: Three representations of the fractional \(p\)-Laplacian: Semigroup, extension and Balakrishnan formulas. Fract. Calc. Appl. Anal. 24(4), 966–1002 (2021). https://doi.org/10.1515/fca-2021-0042
Tisdell, C.C.: On the application of sequential and fixed-point methods to fractional differential equations of arbitrary order. J. Integral Equations Appl. 24(2), 283–319 (2012)
Tuan, N.H., Vo, V.A., Xu, R.: Semilinear Caputo time fractional pseudo-parabolic equations. Commun. Pure Appl. Anal. 20(2), 583–621 (2021)
Vázquez, J.L.: Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete Contin. Dyn. Syst. 7(4), 857–885 (2014)
Vergara, V., Zacher, R.: Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations. J. Evol. Equ. 17, 599–626 (2017)
Vergara, V., Zacher, R.: Optimal decay estimates for time-fractional and other nonlocal subdiffusion equations via energy methods. SIAM J. Math. Anal. 47(1), 210–239 (2015)
Yin, J., Jin., Ch.: Critical extinction and blow-up exponents for fast diffusive p-Laplacian with sources. Math. Meth. Appl. Sci. 30(10), 1147–1167 (2007)
Zacher, R.: Time fractional diffusion equations: solution concepts, regularity, and long-time behavior. in Handbook of Fractional Calculus with Applications. Vol. 2. Fractional Differential Equations, ed. by A. Kochubei and Y. Luchko, De Gruyter, 159–180 (2019)
Acknowledgements
This research has been funded by the Science Committee of the Ministry of Education and Science of the Republic of Kazakhstan (Grant No. AP14972726) and by the FWO Odysseus 1 grant G.0H94.18N: Analysis and Partial Differential Equations. Michael Ruzhansky was supported by the EPSRC grant EP/R003025/2 and by the Methusalem programme of the Ghent University Special Research Fund (BOF)(Grant number 01M01021).
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Borikhanov, M.B., Ruzhansky, M. & Torebek, B.T. Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation. Fract Calc Appl Anal 26, 111–146 (2023). https://doi.org/10.1007/s13540-022-00115-2
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DOI: https://doi.org/10.1007/s13540-022-00115-2