Time-Space Fractional Diffusion Problems: Existence, Decay Estimates and Blow-Up of Solutions

The aim of this paper is to study the following time-space fractional diffusion problem ∂tβu+(-Δ)αu+(-Δ)α∂tβu=λf(x,u)+g(x,t)inΩ×R+,u(x,t)=0in(RN\Ω)×R+,u(x,0)=u0(x)inΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}$$\end{document} where Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^N$$\end{document} is a bounded domain with Lipschitz boundary, (-Δ)α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )^{\alpha }$$\end{document} is the fractional Laplace operator with 0<α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1$$\end{document}, ∂tβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t^{\beta }$$\end{document} is the Riemann-Liouville time fractional derivative with 0<β<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\beta <1$$\end{document}, λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda $$\end{document} is a positive parameter, f:Ω×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} is a continuous function, and g∈L2(0,∞;L2(Ω))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g\in L^2(0,\infty ;L^2(\Omega ))$$\end{document}. Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated.

Here, the order β of Riemann-Liouville fractional operator ∂ β t is defined by ∂ β t u = ∂ t (J 1−β (u − u(0))), where J 1−β denotes the 1 − β order Riemann-Liouville fractional integral operator and it is given by Here Γ is the usual Gamma function. The fractional Laplace operator (−Δ) α , up to a normalization constant, is defined by Here, B ε (x) = {y ∈ R N : |y − x| < ε}. For more properties related to the fractional Laplacian and fractional Sobolev spaces as well as for applications of variational methods to fractional problems, we refer to [3].
The fractional operators and related differential equations have important applications in many areas such as physics [15], mechanics chemistry, population dynamic [4,5], anomalous diffusion [29] and so on. Time fractional differential equations can be used to describe some problems with memory effects. Moreover, both time and space fractional differential equations have been exploited for anomalous diffusion or dispersion where particles spread at a rate inconsistent with Brown motion, see [9]. In the case of time fractional derivatives, particles with "memory effect" propagates slowly, which we call anomalous subdiffusion. Different from the former, spatial fractional diffusion equations are used to describe macroscopic transport and usually result in superdiffusion phenomenon. So far, the works on problems involving the fractional Laplacian and its variants are quite large, here we just list a few, see [8,10,20,21,[31][32][33][34] and the references cited there.
To the best of our knowledge, until recently there has been still very little works on deal with the existence, decay estimates and blow-up of solutions for time-space fractional problems like (1.1). In [30], Vergara and Zacher considered the following time fractional diffusion problem where k * (u − u 0 ) = t 0 k(t − τ )(u(τ ) − u 0 )dτ and k ∈ L 1,loc (R + ). Some useful fundamental identities were obtained. Based on these identities, the existence and Vol. 90 (2022) Time-Space Fractional Diffusion Problems 105 decay estimates of weak solutions were obtained. In particular, the decay estimates of weak solutions were given by using the sub-supersolution method. In [17], Li et al. studied the following time-space fractional Keller-Segel equation x−y |x−y| n−γ+2 ρ(y)dy is the Riesz potential with a singular kernel. The authors obtained the existence and uniqueness of mild solutions. Moreover, the authors discussed the properties of the mild solutions, such as mass conservation and blow-up behaviors. In [1], Bekkai et al. studied the following Cauchy problem involving the Caputo derivative and the fractional Laplacian First the existence of mild solutions of (1.2) was obatined by the Banach contraction mapping principle. Then the authors proved that the mild solution is also the weak solution. Furthermore, the authors showed the local weak solutions blow up in finite time by choosing suitable test function. See also [6,23,36,39] for similar discussions of the blow-up properties of solutions. Very recently, Fu and Zhang [11] considered the following time-space fractional Kirchhoff problem is a continuous function. Under suitable assumptions, the authors obtained the global existence of solutions by using the Galerkin method. Furthermore, a decay estimate of solutions was established. On the other hand, when α, β and s limit to 1, the Eq. (1.1) reduces to the following equation which is called pseudo-parabolic equation. Equations like (1.3) can be used to describe many important physical processes, such as unidirectional propagation of nonlinear, long waves [2,27], the aggregation of population [25] and semiconductors [14]. The study of Eq. (1.3) received much more attention in the past years, see [12,18,35].
Recently, Tuan et al. [28] studied the initial boundary value problem and Cauchy problem of Caputo time-fractional pseudo-parabolic equations where D α t denotes the Caputo time fractional derivative. The local well-posedness of Eq. (1.4) was established. Further, the finite time blow-up of solutions was also obtained. In [24], Nguyen et al. considered a class of pseudoparabolic equations with the nonlocal condition and the Caputo derivative and obtained the existence and uniqueness of the mild solution. In [7], Chaoui and Rezgui dealt with a time fractional pseudoparabolic equation with fractional integral condition. By the Rothe time discretization scheme, the existence of weak solution was obtained. Moreover, the uniqueness of weak solution as well as some regularity results were obtained. Inspired by the above papers, we discuss in this work the existence, uniqueness, decay estimates of weak solutions and solutions that blow up in finite time for problem (1.1) involving the time-space fractional operators. Since our problem is nonlocal, our discussion is more elaborate than the papers in the literature. Comparing with the papers in the literature, the main feature of this paper is that the problem (1.1) contains the Riemann-Liouville time fractional derivative and the fractional Laplacian. Definitely, this paper is the first time to deal with the local existence and global nonexistence of solutions for problems involving the fractional Laplacian and the Riemann-Liouville time fractional derivative.
for any 0 ≤ ϕ ∈ H α 0 (Ω) and a.e. t ∈ (0, T ). u is a weak solution if and only u is both a subsolution and a supsolution. Here, we call u is a global weak solution of problem (1.1), if the equality in above holds for any 0 < T < ∞; u is a local weak solution, if there exists T 0 > 0 such that the equality in Definition 1.1 holds for 0 < T ≤ T 0 .
The following theorem shows the asymptotic behavior of global solutions to problem (1.1).
and f satisfies (f 2 ), then the unique solution of problem (1.1) satisfies the following decay estimates where c 2 > 0 and ϕ 1 > 0 is the eigenfunction corresponding to the first eigenvalue of the fractional Laplacian.
We also discuss the global nonexistence of local solutions for problem (1.1).  In what follows, the letters c, c i , C, C i , i = 1, 2, . . . , denote positive constants which vary from line to line, but are independent of terms that take part in any limit process. Furthermore, for any p ≥ 1 we denote u p = u L p (Ω) .

Preliminaries
In this section, we provide some basic results which will be used in the next sections.
endowed with the norm H α 0 (Ω) is a Hilbert space in which a scalar product is given by the distinct eigenvalues of the fractional Laplace operator and let ω k be the eigenfunction corresponding to λ k of the following eigenvalue problem We obtain for k ∈ N, . The Yosida approximation of the time-fractional derivative operator is an useful tool to deal with problems with Caputo fractional derivative operators. For more details, we refer to [30,37,38]. Let 1 ≤ p < ∞, 0 < β < 1 and X be a real Banach space. Define fractional derivative operator Here, we collect some important properties of g 1−β and B n which are listed in the following: • The kernel g 1−β,n is nonnegative and nonincreasing for all n ∈ N , and g 1−β,n ∈

Remark 1. Obviously, if k is a nonincreasing and nonnegative function in
, for a sufficiently smooth function u, then there holds for a.e. t ∈ (0, T ) Definition 2.4 (see [19]). Let q > 0 be a real number and 0 < T ≤ ∞. We say that a function ω : where R(t) is a continuous and nonnegative function. Clearly, if ω(u) = u r , r > 0, then ω satisfies the condition (q) with any q > 1, and R(t) = e (r−1)qt .
Proof. If β ≥ 1 2 , then the Hölder inequality implies that which yields the desired result. Now we consider the case 0 < β < 1/2. By the Hölder inequality, we have which ends the proof.

Remark 2. The Caputo derivative of an absolutely continuous function u is defined as follows
In view of the definition of the Riemann-Liouville derivative, we know that the Riemann-Liouville derivative and the Caputo derivative have the relationship

Existence and Uniqueness of Weak Solutions
In this section by means of the Galerkin method, we establish the existence of local solutions to the problem (1.1). Assume that {ω k } is an orthonormal basis in L 2 (Ω) and then we shall find Galerkin approximation solutions u m = u m (t) of the following form where a mj satisfies that for j = 1, 2, . . . , m. Here, (·, ·) denotes the inner product of L 2 (Ω). Problem (3.1) is a nonlinear fractional ordinary differential system. Next, we show that problem (3.1) has a unique local solution for every m ∈ N . First, by Lemmas 2.2 and 2.3, we give a prior estimate for problem (3.1).
Proof. Multiplying (3.1) by ∂ β t a mj and summing j from 1 to m, we have By the Yosida approximation of time Riemamm-Liouville fractional derivative, Lemma 2.2 and Hölder's inequality, one can deduce that . By using the Hölder inequality and Young inequality, we deduce and S * is the embedding constant from H α 0 (Ω) to L 2 * α (Ω). Here, we have used the fact that (p − 1)q ≤ 2 * α , thanks to p ≤ 2 * α . We also get Choose ε small enough such that S * λε < 1. Then it follows from (3.4) that Convolving (3.5) with g β and letting n go to ∞ and selecting an appropriate subsequence(if necessary), it leads to where > 0. Then we deduce from (3.6) that . Considering the definition of Ω −1 and using Ω(2a 2 (t)) + g 1 (t) ∈ DomΩ −1 , we get which implies that for all 0 < t < T 1 . When β ∈ (0, 1 2 ), R(t) = e (p−2)(z+2)t , then we have Consider the definition of Ω −1 , we obtain In conclusion, there exist T * = min{T 1 , T 2 } and C > 0 such that u m H α 0 (Ω) ≤ C for all 0 < t < T * .
Based on Lemmas 3.1 and 3.2, we obtain the following estimate.
Integrating above inequality from 0 to T and letting n → ∞, we obtain For the case f satisfies the Lipschitz condition, by Lemma 3.2 and a similar discussion as above, one can obtain the desired result. Now, we prove the local existence of solutions for system (3.1).

Theorem 3.4. Under the assumptions of Theorem 1.2, system (3.1) has a unique solution for all
Proof. First, problem (3.1) is equivalent to the problem where ψ(t) = (a mj (t)) ∈ R m ,A = diag( Laplace transform or convoluting with g β , we transform (3.7) into the following Volterra type system Therefore, we only need to prove that system (3.8) admits a unique continuous solution. Notice Then we obtain a prior estimate ψ(t) R m ≤ R 0 .
Consequently, we prove that Φ is contractive on E T provided T is small enough such that DT 1−β < 1. Thus, by the Banach contraction mapping theorem, we know that the map Φ has a unique fixed point on some small interval [0, T 0 ]. Therefore, we prove that system (3.7) has a unique solution on [0, T 0 ]. On the other hand, if f satisfies the Lipschitz condition, the existence of unique solution of system (3.7) on some small interval [0, T 0 ] can be proved similarly as above.
Finally, we show that the local solution can be extended to (0, T ]. Let T 0 and u m (T 0 ) be the initial data. Then repeating the same process as above, we can get a unique continuous solution on [T 0 , for all v ∈ C 1 (0, T ; H α 0 (Ω)). Since {∂ β t u m } is bounded in L 2 (0, T ; L 2 (Ω)), up to a subsequence we may assume that ∂ β t u m χ in L 2 (0, T ; L 2 (Ω)).