Abstract
In this paper, we discuss an initial value problem for the semilinear time-fractional diffusion equation. The local well-posedness (existence and regularity) is presented when the source term satisfies a global Lipschitz condition. The unique continuation of solution and finite time blowup result are presented when the reaction terms are logarithmic functions (local Lipschitz types).
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1 Introduction
Let \(\Omega \subset {\mathbb {R}}^N, ~ ( N \ge 1)\) be a bounded open set with boundary \(\Omega ^c\). The main aim of the paper is to study the properties of the solutions of a class of time-fractional diffusion equations involving the so-called Riemann–Liouville (R–L) time-fractional derivative. More precisely, we consider the following initial value problem:
where \(T>0\), \(\alpha \in (0,1)\) is real number and \(\dfrac{\partial ^{1-\alpha } }{\partial t} u \) denotes the R–L time-fractional derivative of order \(1-\alpha \) of the function u formally given by
where the Riemann–Liouville fractional integral operator \(\mathcal J^\alpha : L^2(0,T) \rightarrow L^2(0,T)\) is defined by the formula (see, e.g., [1])
and \(\Gamma (\cdot )\) is the Gamma function. The operator \({\mathcal {A}}\) is a linear, positive definite, self-adjoint operator with compact inverse in \(L^2(\Omega )\), \(u = u(x, t)\) is the state of the unknown function and \(u_0(x)\) is a given function. The function F is a nonlinear source term which appears in some physical phenomena [2,3,4].
When \(\alpha = 1\) and \({\mathcal {A}} = -\Delta \) problem \({{\mathbb {P}}}\) describes the nonlinear heat Eq. [2, 5,6,7]
If \(\alpha \in (0,1)\), Problem \({{\mathbb {P}}}\) is called an initial value problem for the semilinear time-fractional diffusion equation; we refer the reader to [3, 8,9,10] and the references therein. Many important physical models and practical problems require one to consider the diffusion model with a fractional derivative rather than a classical one, like physical models considering memory effects [2,3,4, 11,12,13] and some corresponding engineering problems [2, 3, 14, 15] with power-law memory (non-local effects) in time [4, 8, 16,17,18,19,20]. For nonlinearities of power-type \(F(u)=|u|^{p-1}u\) for \(p \ge 1\), Bruno de Andrade et al. [3] considered the fractional reaction–diffusion equation to discuss the global well-posedness and asymptotic behavior of solutions; see also [7, 21] and the references therein. Studies of logarithmic nonlinearity have a long history in physics as they occur naturally in inflation cosmology, quantum mechanics, and nuclear physics [22] and PDEs with logarithmic nonlinearity have attracted many authors; see [23,24,25,26] and the references therein.
Results on initial value problems for R–L time-fractional diffusion equation with logarithmic nonlinearity are quite limited. The solution operator of our problem \(E_{\alpha ,1}\left( - {\mathcal {A}} t^{\alpha } \right) \) brings some difficulties in estimating and analyzing the solution (existence and regularity estimate of the solutions). We consider the model with the source terms \(F_p(u) = \eta V_p(u) \log |u|\) and \(V_p(u) = |u|^{p-2} u, ~ p\ge 2, \eta >0\) (locally Lipschitz type). To present the properties of the solutions in \(W^{s,q}(\Omega )\), we need to consider the Lipschitz properties of the source function (both global Lipschitz property and local Lipschitz property). Based on the conditions of the constants s, q depending on the dimensions \(N \ge 1\) and the constant \(s>0\), we set up the Sobolev embeddings \({\mathbb {X}}^{s}(\Omega ) \hookrightarrow W^{s,q}(\Omega ) \hookrightarrow L^p(\Omega )\) (see the definition of the spaces \(W^{s,q}(\Omega )\) and \({\mathbb {X}}^{s}(\Omega )\) in (2.3) and (2.7) below).
In Sect. 2, we present some basic definitions and the setting for our work. Moreover, we obtain a precise representation of solutions using Mittag–Leffler operators. In Sect. 3, we first present local well-posedness results when the source term satisfies a global Lipschitz condition. Also local existence, continuation of solutions and finite time blowup results are presented when the source terms are logarithmic functions.
2 Notations and preliminaries
2.1 Relevant notations and the functional spaces
Given two positive quantities y, z, we write \(y \lesssim z\) if there exists a constant \(C > 0\) such that \(y \le Cz\). Let us recall that the spectral problem
admits a family of eigenvalues
Given a Banach space B, let C([0, T]; B) be the set of all continuous functions which map [0, T] into B. The norm of the function space \(C^k ([0, T ]; B),\) for \(0 \le k \le \infty \) is denoted by
For any real numbers \(s>0\) and \(1 \le p < \infty \), we recall the fractional Sobolev-type spaces \(W^{s,p}(\Omega )\) via the Gagliardo approach (also called Aronszajn or Slobodeckij spaces). Fix a number \(s \in (0,1)\) and for any \(p \in [1,\infty )\), define \(W^{s,p}(\Omega )\) as follows
For \(0<s<1\), it can be said that \(W^{s,p}(\Omega )\) is an intermediate Banach space between \(L^p(\Omega )\) and \(W^{1,p}(\Omega )\), endowed the corresponding norm
where the seminorm
denotes the Gagliardo (semi)norm of v. For \(p=2\) in (2.3), together with the norm \(\left\| \cdot \right\| _{W^{s,2}(\Omega )}\) the space becomes a Hilbert space. Let us also set \( W_0^{s,2} (\Omega ) = \overline{C_c^\infty (\Omega )}^{W^{s,2} (\Omega )}.\) It is well known that if \(\Omega \) is bounded, then we have the following continuous embeddings:
for more details on fractional Sobolev spaces see [17, 27] and the references therein.
For each number \(s\ge 0\), we define
Let us denote by \(H^s(\Omega )\) the Sobolev–Slobodecki space \(W^{s,p}(\Omega )\) when \(p=2\), and by \(H^s_0(\Omega )\) the closure of \(C_c^\infty (\Omega )\) in \(H^s(\Omega )\). Throughout this paper, \(\Omega \) is assumed to be smooth enough such that \(C_c^\infty ( \Omega )\) is dense in \(H^s( \Omega )\) for \(0<s< \frac{1}{2}\). This guarantees \(H_0^s(\Omega )=H^{s}(\Omega )\). Moreover, it is well-known that
where we denote by \(H_{00}^{1/2} ( \Omega )\) the Lions–Magenes space. Let \({\mathbb {X}} ^{-s}( \Omega )\) be the duality of \( {\mathbb {X}} ^{s} \) which corresponds to the dual inner product \(\left( \cdot ,\cdot \right) _{-s,s}\). Then, the operator \({\mathcal {A}}^s: {\mathbb {X}}^{s}( \Omega ) \rightarrow {\mathbb {X}} ^{-s} (\Omega ) \) of the fractional power s can be defined by
The above settings can be found in [28] (Sect. 3) and [29] (Sect. 2). In the next lemmas, we present some useful embeddings between the spaces mentioned above.
Lemma 2.1
Given \(1 \le p , p' < \infty \), \(0 \le s \le s' <\infty \) and \(s' - \frac{N}{p'} \ge s - \frac{N}{p}\). Then
Lemma 2.2
Let \(0\le s \le s' \le 2\) and let \(H^{-s}( \Omega )\) be the dual space of \(H_0^s( \Omega )\). Then the following embeddings hold
and
2.2 Properties of Mittag–Leffler functions and some related results
The Mittag–Leffler function is defined by (see [30])
where \(\alpha >0\) and \(\alpha ' \in {\mathbb {R}}\) are arbitrary constants, \(\varGamma \) is the usual gamma function.
Next, we give some properties of the Mittag–Leffler function. Let \(\alpha '\in {\mathbb {R}},\) and \(\alpha \in (0,2)\), we have:
where \(C>0\) depends on \(\alpha , \alpha ',\tau \) and \(\frac{\pi \alpha '}{2}< \tau < \min \{\pi ,\pi \alpha '\}\) (see e.g. [30]).
Lemma 2.3
(See [30, 31]) For \(0< \alpha _1<\alpha _2 <1\) and \(\alpha \in [\alpha _1,\alpha _2],\) there exist positive constants \({\underline{C}}, {\overline{C}},\) such that
Lemma 2.4
(See [31]) Let \(\alpha , \lambda , \gamma \) are positive constants, and for every \(t>0, n \in {\mathbb {N}}\), we have
Lemma 2.5
(See [32]) The following equality holds
where we recall the definition of the Wright-type function
Moreover, \({\mathcal {M}}_\alpha (s)\) is a probability density function, that is,
Lemma 2.6
(See [3], expression (6), for \({\mathcal {A}} = -\Delta \)) The function u is a mild solution of \({{\mathbb {P}}}\) if \(u \in C([0, T];L^2(\Omega ))\) and satisfies the following integral equation
for all \(t <T\), and \(\alpha \in (0,1)\).
Lemma 2.7
(a) (Weakly singular Grönwall’s inequality, see [33], Theorem 1.2, page 2) Let \(a,b, \beta , \beta '\) be non-negative constants and \(\beta , \beta ' <1\). Assume that \(\varphi \in L^1[0,T]\) satisfies
Then there exists a constant \(C(b, \beta ', T )\) such that
(b) (Fractional Grönwall’s inequality, see [34], Corollary 2) Assume \(\beta >0\), \(\varphi \) is nonnegative, locally integrable, and
on (0, T), where a, b are positive constants. Then,
Lemma 2.8
(a) For \(z >0\), then there exists a constant \(C>0\) depending on \(\theta \) such that
(b) (Hölder’s inequality for negative exponents) (see [35]) Let \(k'<0\), and \(k \in {\mathbb {R}}\) be such that \(\frac{1}{k'} + \frac{1}{k} = 1\) and \(f(x),g(x) \ge 0, ~ \forall x \in \Omega \) are Lebesgue measurable functions. Then
Proof
The proof of inequalities (2.20) and (2.21) are elementary, so we omit them here. \(\square \)
Lemma 2.9
(See [17, 27]) Let \(\Omega \subset {\mathbb {R}}^N\), \(k,m \in {\mathbb {N}}\) with \(k \ge m\) satisfying \((k-m)p <N\) and \(1\le p <\infty \). Then we have the following Sobolev embeddings
where \(p^*_{k,m}, 2^*_s\) are the so-called fractional Sobolev exponents, given by
3 Main results
3.1 The case when the source terms are globally Lipchitz functions
In this section, we will study the existence and uniqueness of mild solutions to problem \({{\mathbb {P}}}\). First we assume the global Lipschitz continuity and the time Hölder continuity on the nonlinear term. More precisely, we suppose that \(F:{\mathbb {X}}^{p}(\Omega ) \rightarrow {\mathbb {X}}^{q}(\Omega )\), \(F(0)=0\), and
where \(K:[0,T] \rightarrow {\mathbb {R}}_+\) and p, q are real numbers.
Our results in this section present the local well-posedness of the problem. Here, \( {\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^q(\Omega ))\) denotes the weighted space of all functions \(v\in C ((0,T];{\mathbb {X}}^q(\Omega ))\) such that
where \(\beta>0,~d>0\). First we state the following lemma which will be useful in our main results. (This lemma can be found in [36], Lemma 8, page 9.)
Lemma 3.1
Let \(a>-1\), \(b>-1\) such that \(a+b \ge -1\), \(h>0\) and \(t\in [0,T]\). For \(\mu >0\), the following limit holds
Now, we are in the position to introduce the main contributions of this work. Our main results address the existence and regularity of the mild solution.
Theorem 3.1
Let \(0<\beta <1 \). Assume that \(q-p <\min \left\{ \frac{2(1-\beta )}{\alpha },\frac{2\beta }{\alpha } \right\} \). Let \(u_0 \in {\mathbb {X}}^{q-2\gamma }(\Omega )) \) for any \(0<\gamma <\min \left\{ \frac{\beta }{\alpha };1\right\} \). Then Problem \({{\mathbb {P}}}\) has a unique solution u in \({\mathbb {Z}}_{\beta ,d_0}((0,T];{\mathbb {X}}^q(\Omega ))\) with some \(d_0>0\). Moreover, there exist positive constant C independently of t, x and for \(1/2<\beta <1\), \(1-\beta<\alpha <1/2\) such that
Proof
Define the mapping \({\mathfrak {B}}: \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) \rightarrow \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) \), \(d>0\), by
In what follows, we shall prove the existence of a unique solution of Problem \({{\mathbb {P}}}\). This is based on the Banach principal argument. First, since \(0<\gamma <1\), we have
It follows from the condition \(\beta >\alpha \gamma \) that
From the latter inequality, we deduce that \(u_0 \in \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega ))\). Indeed, for \(w_1,w_2\in {\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) \), we have
We derive the estimate
where
From the conditions of \(\alpha ,\beta , p, q\), we find that
Applying Lemma 3.1, we obtain that
Hence, there exists a positive \(d>0\) such that \({\mathfrak {B}}\) is a contraction mapping on \(\mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega )) \). This together with (3.5) leads to \({\mathfrak {B}} w \in \mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega )) \) if \(w \in \mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega )) \). Hence, we conclude that \({\mathfrak {B}}\) has a fixed point u in \(\mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega )) \), i.e, u is a unique mild solution of Problem \({{\mathbb {P}}}\).
This and the technique in (3.6) yields
Multiplying both sides to \(t^{\beta } e^{-dt}\), we find that
By applying the Hölder inequality, and then using \(e^{-2d(t-\tau )}<1\), we can find some positive constant \({\mathscr {M}}\) such that
Taking the estimate (3.8), and (3.10) together gives that
where
Applying Lemma 2.7(b), we deduce that
The proof of Theorem 3.1 is completed. \(\square \)
3.2 The case when the source terms are locally Lipschitz functions
Next, we shall present the results when the source terms are logarithmic nonlinearities of the following type \(F_{p}(u) = \eta V_{p}(u) \log |u|\) and \(V_p(u) = |u|^{p-2}u\), for \(p \ge 2, \eta >0\).
Remark 3.1
For the source terms of polynomial type nonlinearities, i.e., \(F_p(u) = V_p(u)\) a simpler result was considered in [2, 5].
Lemma 3.2
For \(F_p(u)(x,t) = \eta V_p(u)\log |u| \in L^\infty (\Omega \times (0,T) \times {\mathbb {R}}), ~ p\ge 2, \eta >0\), there exists a positive constant C such that
for all \((x,t) \in \Omega \times (0,T),~ \forall u, w \in \mathbb R\).
Proof
For \((x,t) \in \Omega \times (0,T)\) and \(u,w \in {\mathbb {R}}\), we have
Thanks to the results in [5], we have that
Using the basic inequality \(\log (1+z) < z\) for \(z>0\), one has
From (3.14)–(3.16), we have the proof of Lemma 3.2. \(\square \)
Theorem 3.2
(Local existence) Let \(\alpha \in \left( 0, 1\right) \), \(N\ge 1, p\ge 2,0\le s < s_2\), for \(0\le s_2 <N/2\). Let \(1 \le q \le \min \left\{ 2^\star _{s_2,s};\frac{N \theta }{N+\theta s} \right\} \) with \(2^\star _{s_2,s}\) satisfying \(\frac{1}{2^\star _{s_2,s}} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N}\) and \(qs < N\). Let \(u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )\), and for the nonlinearity source of logarithmic function type
then there is a time constant \(T>0\) (depending only on \(u_0\)) such that Problem \({{\mathbb {P}}}\) has a unique mild solution belonging to \( C([0,T];W^{s,q}(\Omega ))\).
Remark 3.2
In Theorem 3.2, for \(N\ge 1\), and \(0\le s_2 <N/2\) let us choose \(N = 3, s_2 =1\). From the conditions
Then, the Problem \({{\mathbb {P}}}\) has a unique mild solution \(u \in C([0,T]; L^{q}(\Omega )), ~ 1 \le q \le 6\), or \(u \in C([0,T]; W^{1,q}(\Omega ))\), for \(1 \le q \le 2\).
Proof
For \(N\ge 1, p\ge 2\), \(0< \theta \le p-1\) (\(\theta \) is defined in Lemma 2.8), we put
where \(2^*_{s_2,s} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N},\) and Z(a, b) be defined by
with the pairs (a, b) as follows:
Let \(T>0\) and \(R>0\) to be chosen later, and we consider the following space
for \(0<\alpha <1\), and we define the mapping \({\mathbf {M}}\) on \(\mathbb W\) by
We show that \({\mathbf {M}}\) is invariant in \({\mathbb {W}}\) and \({\mathbf {M}}\) is a contraction.
\(\bullet \) Claim I If \(u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )\), then \({\mathbf {M}}\) is \({\mathbb {W}}\)-invariant. In fact, from Lemma 2.3(b), we have
From (3.19), one has \(s < s_2\) and \( 1\le q \le 2^*_{s_2,s}\), and we have that \({\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow H^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\) and then, we conclude from (3.24) that
From (3.20), we have \(ps_2< s^* < 1+ (p-1)s_2\), this implies that \(0< s_2 - s_1 <1\) and for \(ps_2< s^* < p s_2 + \frac{N}{2},~\) or \(~- \frac{N}{2}< p s_2 - s^* < 0\) thus \(-\frac{N}{2}< s_1 < 0.\) Taking \(\frac{1}{2^*_{s_1} } = \frac{1}{2} - \frac{s_1}{N}\) and combine with Lemma 2.9, we obtain \(L^{2^*_{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega )\). Using Lemma 2.3(b), we have for \(t \in (0,T]\)
Let us set \(\Omega ^-:= \{x\in \Omega : |u(x)| < 1\}\) and \(\Omega ^+:= \{x\in \Omega : |u(x)| \ge 1\}\). Using Hölder’s inequality, we have
where we have used the elementary inequality \((a+b)^c \le a^c+b^c\), for \(0<c<1\). From the inequality (2.20) for \(|u(x)|<1, ~ \forall x \in \Omega \), by applying Lemma 2.8b) for \(k' = -\frac{1}{p2_{s_1}^*} <0\), we have
From the inequality (2.20) for \(|u(x)| \ge 1\), we have
From (3.27), (3.28) and (3.29), we conclude that
For \(s>0, qs <N\), and \(q < \frac{N\theta }{N+\theta s}\), this implies that \(q^*_s \le \theta \) with \(q^*_s\) satisfies
we deduce from Lemma 2.9 that the following Sobolev embedding holds \(L^{\theta }(\Omega ) \hookrightarrow W^{s,q}(\Omega )\). Then we get that
and for \(\theta >0\), we have that
From (3.21), the constant \(s^* > Z(1,1),\) [for Z(1, 1) defined in (3.21)] and observe that
then we also obtain \(W^{s,q}(\Omega ) \hookrightarrow L^{p2^*_{s_1}}(\Omega )\). For \(s^* > Z(\theta ,1)\) [for \(Z(\theta ,1)\) defined in (3.21)], we infer that
this implies that \(W^{s,q}(\Omega ) \hookrightarrow L^{p \theta 2^*_{s_1} }(\Omega )\). This implies that
and from (3.26), and for \(0< \theta <p-1\), we have
where from (3.22), we have that \(\big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )} \le R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )}\), for all \(\tau \in [0,T]\). From (3.26), (3.32), we obtain that for \(t \in (0,T]\)
For the constants s, q satisfying (3.19), we have that \({\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\) and \(\alpha \in (0,1)\), and for all \(t \in [0,T]\), we get
Hence, from (3.24) and (3.34), for every \(t\in (0, T]\),
Therefore we see that if \( R = 2 C \left\| u_0\right\| _{\mathbb X^{s_2}(\Omega )} \) and for the constant \(C>0\), \(\theta < p-1\) such that
and
Then, we imply \({\mathbf {M}}\) is invariant in \({\mathbb {W}}\).
\(\bullet \) Claim II \({\mathbf {M}}: {\mathbb {W}} \rightarrow {\mathbb {W}}\) is a contraction map. Let \(u,w\in {\mathbb {W}}\), and similar to (3.26) and using Lemma 2.3(b), one has for every \(t\in (0,T]\),
in which we used the Sobolev embedding \(L^{2^*_{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega )\), for \(\frac{1}{2^*_{s_1}} = \frac{1}{2} - \frac{s_1}{N}\) and \(-\frac{N}{2} < s_1 \le 0\). By recalling Lemma 3.2, we arrive at
For the constant \(2^*_{s_1} \ge 1\), using Hölder’s inequality, we get
From the inequality (2.20) for \(|u(x)|<1, ~ \forall x \in \Omega \), we have
where we have chosen \(k' = -\frac{p-1}{p2^*_{s_1}} <0\) in Lemma 2.8(b). For \(|u(x)| \ge 1, ~ \forall x \in \Omega \), we have
From (3.38), (3.39) and (3.40), we conclude that
Thanks to Hölder’s inequality, we get that
Similar to (3.39) and (3.40), we have the following estimate
Combining (3.42) and (3.43), we get that
Similarly,
We use the Hölder’s inequality to obtain that
Combining the results obtained in (3.37), (3.41), (3.44), (3.45) and (3.46), we have
From (3.18)–(3.21) we have the following:
- \(\triangleright \):
-
For \(q^*_s\) satisfying \(\frac{1}{q^*_s} = \frac{1}{q} - \frac{s}{N}\), for \(q \le \frac{N\theta }{N+\theta s}\) and \(sq < N\), then \(q^*_s \le \theta \) and we deduce from Lemma 2.9 that \(L^{\theta }(\Omega ) \hookrightarrow W^{s,q}(\Omega )\). This implies that
$$\begin{aligned} \big \Vert u\big \Vert _{L^{\theta }(\Omega )}^{-\theta } \le C \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{-\theta }, \quad \hbox {for}~~ 0< \theta \le p-1. \end{aligned}$$ - \(\triangleright \):
-
For \(s^* > Z(1,1)\), a similar argument with (3.31) and we observe that \( q^*_s \ge p 2^*_{s_1},\) and then we deduce from Lemma 2.9 that the following Sobolev embedding holds \(W^{s,q}(\Omega ) \hookrightarrow L^{p2^*_{s_1}}(\Omega )\).
- \(\triangleright \):
-
For \(s^*> Z(p-2,1)\), implies that
$$\begin{aligned} p(p-2) 2^*_{s_1} = \frac{2p(p-2)N}{N-2s_1} < \frac{2p(p-2)N}{N+2 Z(p-2,1) -2 p s_2} \le q^*_s, \end{aligned}$$(3.48)and we infer that \(W^{s,q}(\Omega ) \hookrightarrow L^{p (p-2) 2^*_{s_1}}(\Omega )\).
- \(\triangleright \):
-
For \(s^*> Z(\theta ,p-1)\), we observe that \(\frac{p \theta 2^*_{s_1}}{p-1} \le q^*_s,\) then we get \(W^{s,q}(\Omega ) \hookrightarrow L^{\frac{p \theta 2^*_{s_1}}{p-1}}(\Omega )\).
- \(\triangleright \):
-
For \(s^*> Z(\theta ,p-2)\), we have \(\frac{p \theta 2^*_{s_1}}{p-2} \le q_s^*,\) and this implies that \(W^{s,q} (\Omega ) \hookrightarrow L^{\frac{p \theta 2^*_{s_1}}{p-2}}(\Omega )\).
- \(\triangleright \):
-
For \(s^*> Z(p-2,p-1)\), implies \(\frac{p (p-2) 2^*_{s_1}}{p-1} \le q^*_s,\) and we infer that \(W^{s,q}(\Omega ) \hookrightarrow L^{\frac{p (p-2) 2^*_{s_1} }{p-1}}(\Omega )\).
We can now combine the results above together with (3.47) to deduce that
for all \(t\in (0,T]\), we have used that \(\max \left\{ \Vert u\Vert _{W^{s,q}(\Omega )}; \Vert w\Vert _{W^{s,q}(\Omega )} \right\} \le R + \Vert u_0\Vert _{W^{s,q}(\Omega )}\), and for the constant \(K(R,u_0):=K(N,p,\theta ,s_1,R, \Vert u_0\Vert _{W^{s,q}(\Omega )})\) but independent of t. From this, one observes that
Inserting the result of (3.50) into (3.36), we obtain that
For the constants s, q satisfying (3.19), we have that \({\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\), and
Choosing \(T, K(R,u_0)\) small enough such that \(CK(R,u_0) T^{1-\alpha } < 1,\) it follows that \({\mathbf {M}}\) is a contraction map on \(\mathbb W\). So, we invoke the contraction mapping principle to conclude that the map \({\mathbf {M}}\) has a unique fixed point u in \({\mathbb {W}}\). The proof of Theorem 3.2 is completed. \(\square \)
Since we already know that the mild solution of \({{\mathbb {P}}}\) does exist, the question is whether it will continue (continuation to a bigger interval of existence) and in what situation it is non-continuation by blowup.
Definition 3.1
Given a mild solution \(u \in C([0,T]; W^{s,q}(\Omega ))\) of \({{\mathbb {P}}}\) for \(\alpha \in (0,1)\), we say that \(u^\star \) is a continuation of u in \((0, T^\star ]\) for \( T^\star > T\) if it is satisfies
Theorem 3.3
(Continuation) Suppose that the assumptions of Theorem 3.2 are satisfied. Then, the mild solution (unique) on (0, T] of Problem \({{\mathbb {P}}}\) can be extended to the interval \((0, T^\star ],\) for some \( T^\star > T \), so that, the extended function is also the mild solution (unique) of Problem \({{\mathbb {P}}}\) on \((0, T^\star ].\)
Proof
Let \(u:[0, T] \rightarrow W^{s,q}(\Omega )\) be a mild solution of Problem \({{\mathbb {P}}}\) (T is the time from Theorem 3.2). Fix \(R^\star >0,\) and for \( T^\star > T,\) (\(T^\star \) depending on \(R^\star \)), we shall prove that \(u^\star : [0,T^\star ] \rightarrow W^{s,q}(\Omega )\) is a mild solution of Problem \({{\mathbb {P}}}\). Assume the following estimates hold:
where \(0<\theta \le p-1\) and \(K(R,u_0)\) is defined in the proof of Theorem 3.2. For \( T^\star \ge T >0\) and \(R^\star >0\), let us define
\(\bullet \) Step I We show that \({\mathbf {M}}\) defined as in (3.23) is the operator on \({\mathbb {W}}^\star \). Let \(u^\star \in {\mathbb {W}}^\star \) and we consider two cases.
\(*\) If \(t\in (0,T]\), then by virtue of Theorem 3.2, we have the Problem \({{\mathbb {P}}}\) has a unique solution and we also have \(u^\star (\cdot ,t)=u(\cdot ,t)\). Thus \({\mathbf {M}} u^\star (t) = {\mathbf {M}} u(t) = u(\cdot ,t) \) for all \(t \in (0, T]\).
\(*\) If \(t\in [ T, T^\star ]\), we have
Estimating the term \(\left\| J_2(u_0)( t)\right\| _{W^{s,q}(\Omega )}\), using Lemma 2.5, we have for all \(t \in [T,T^\star ]\),
For \(z>0\), using the inequality \(1-e^{-z}\le z, ~ \hbox { and} ~~ ze^{-z} \le 1,\) one obtains
where we have use the inequalities
For the constants s, q satisfying (3.19), we have that \(\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\). Hence, we get that
From (3.53), this implies that the following estimate holds
Similar to (3.32), we have the following estimate for all \(t \in [T,T^\star ]\) (note that we can choose \(T^\star >T\) and close enough to T)
where from (3.59), for all \(t \in [T,T^\star ]\), we have used that
Using (3.54) and (3.55), we infer that
We continue with the estimate on the third term of (3.60), and using Lemma 2.3(b) and Lemma 2.4, we obtain for all \(t \in [ T, T^\star ]\)
For the constant \(s_1\) satisfying \(-\frac{N}{2} < s_1 \le 0\) and \(\frac{1}{2^\star _{s_1} } = \frac{1}{2} - \frac{s_1}{N}\), from Lemma 2.9, we obtain \(L^{2^\star _{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega ).\) Hence, we deduce that
In the same way as in (3.32), and from (3.59), one obtains
for \(\theta < p-1\). From (3.67), we have that \(\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\) and obtain
Thus, for \(t\in (T,T^\star ]\), we obtain
From (3.56) and (3.57), we get
It follows from (3.63), (3.65), (3.71) that, for every \(t\in [ T, T^\star ]\)
We have shown that \({\mathbf {M}}\) is a map \({\mathbb {W}}^\star \) into \({\mathbb {W}}^\star \).
\(\bullet \) Step II We show that \({\mathbf {M}}\) is a contraction on \({\mathbb {W}}^\star \). Let \( u,w \in {\mathbb {W}}^\star \), and we have that for \( 0 \le t \le T^\star ,\)
where we note that \({\mathbf {M}}{\mathbf {u}}(t)-{\mathbf {M}}{\mathbf {w}}(t) = 0,\) vanishes in \({\mathbb {W}}^\star \) for all \(t \in (0, T]\). Then, for all \(t \in [0, T^\star ]\), proceeding as in Claim (2) of the last theorem, we have
Thus, using the Sobolev embedding \({\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\) with s, q satisfying (3.19), for all \(T^\star >0\), so without loss of generality, we may assume that \(0\le R^\star <4\), and we infer that
This implies that \({\mathbf {M}}\) is a \(\frac{R^\star }{4}\)-contraction. By the Banach contraction principle it follows that \({\mathbf {M}}\) has a unique fixed point \(u^\star \) of \({\mathbf {M}}\) in \({\mathbb {W}}^\star \), which is a continuation of u. This finishes the proof. \(\square \)
The next results are on global existence or non-continuation by a blowup.
Definition 3.2
Let u(x, t) be a solution of \({{\mathbb {P}}}\). We define the maximal existence time \( T_{\max }\) of u(x, t) as follows:
-
(i)
If u(x, t) exists for all \(0 \le t < \infty \), then \( T_{\max } = \infty \).
-
(ii)
If there exists \( T \in (0,\infty )\) such that u(x, t) exists for \(0 \le t < T\), but does not exist at \(t = T\), then \( T_{\max }= T\).
Definition 3.3
Let u(x, t) be a solution of \({{\mathbb {P}}}\). We say u(x, t) blows up in finite time if the maximal existence time \( T_{\max }\) is finite and
Theorem 3.4
(Global existence or finite time blowup) For \(N\ge 1, p\ge 2,0\le s < 2s_2\), for \(0\le s_2 <N/2\) and \(1 \le q \le \min \left\{ 2^\star _{s_2,s};\frac{N \theta }{N+\theta s} \right\} \) with \(2^\star _{s_2,s}\) satisfying \(\frac{1}{2^\star _{s_2,s}} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N}\) and \(qs < N\). For \(u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )\), there exists a maximal time \(T_{\max } >0\) such that \(u \in C([0,T_{\max }];W^{s,q}(\Omega ))\) is the mild solution of \({{\mathbb {P}}}\). Thus, either Problem \({{\mathbb {P}}}\) has a unique global mild solution on \([0,\infty )\) or there exists a maximal time \(T_{\max } < \infty \) such that
Proof
Let \(u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )\) and define
Assume that \(T_{\max } < \infty \), and \(\Vert u(\cdot ,t) \Vert _{\mathbb X^{s_2}(\Omega )} \le R_0,\) for some \(R_0 >0\). Now suppose there exists a sequence \(\{t_{n}\}_{n\in {\mathbb {N}}}\subset [0, T_{\max })\) such that \(t_n \rightarrow T_{\max }\) and \(\{u(\cdot ,t_{n})\}_{n\in {\mathbb {N}}} \subset \mathbb X^{s_2}(\Omega )\). Let us show that \(\{u(\cdot ,t_{n})\}_{n\in \mathbb N}\) is a Cauchy sequence in \({\mathbb {X}}^{s_2}(\Omega )\). Indeed, given \(\epsilon >0\), fix \(N \in {\mathbb {N}}\) such that for all \(n,m>N\), \(0< t_n< t_{m} < T_{\max }\), we have
Similar to (3.61), and using the Sobolev embedding \(\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )\), we have that
In the same way as in (3.32), we get
Similar to (3.68), we have
and
Thus, since \(\{t_n\}_{n\in {\mathbb {N}}^*}\) is convergent we can take \(N:=N(\epsilon ) \in {\mathbb {N}}^*\) with \(m \ge n \ge N\) such that \(|t_{m} - t_n|\) is as small as we want, and we have
and
Hence, given \(\epsilon >0\) there exists \(N \in {\mathbb {N}}\) such that
It follows that \(\{u(\cdot ,t_n)\}_{n\in {{\mathbb {N}}}} \subset W^{s,q}(\Omega )\) is a Cauchy sequences and for \(\{t_n\}_{n\in {\mathbb {N}}^*}\) arbitrary we have proved the existence of the limit
FRom our previos result we deduce that the solution can extended to some larger interval (u can be continued beyond \( T_{\max }\)), and this contradict the definition of \( T_{\max }\). Thus, either \( T_{\max } = \infty \) or if \( T_{\max }<\infty \) then \(\lim _{t\rightarrow T_{\max }^-} \left\| u(\cdot ,t)\right\| _{W^{s,q}(\Omega )} = \infty \). The proof of Theorem 3.4 is finished. \(\square \)
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Acknowledgements
Vo Van Au and Nguyen Huy Tuan were supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.02-2019.09. Bruno de Andrade is partially supported by CNPQ, Brazil under Grant Number 308931/2017-3 and CAPES, Brazil under Grant Number 88881.157450/2017-01.
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de Andrade, B., Van Au, V., O’Regan, D. et al. Well-posedness results for a class of semilinear time-fractional diffusion equations. Z. Angew. Math. Phys. 71, 161 (2020). https://doi.org/10.1007/s00033-020-01348-y
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DOI: https://doi.org/10.1007/s00033-020-01348-y