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

Abstract

AbstractThe aim of this paper is to study the following time-space fractional diffusion problem $$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}$$ ∂ t β u + ( - Δ ) α u + ( - Δ ) α ∂ t β u = λ f ( x , u ) + g ( x , t ) in Ω × R + , u ( x , t ) = 0 in ( R N \ Ω ) × R + , u ( x , 0 ) = u 0 ( x ) in Ω , where $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is a bounded domain with Lipschitz boundary, $$(-\Delta )^{\alpha }$$ ( - Δ ) α is the fractional Laplace operator with $$0<\alpha <1$$ 0 < α < 1 , $$\partial _t^{\beta }$$ ∂ t β is the Riemann-Liouville time fractional derivative with $$0<\beta <1$$ 0 < β < 1 , $$\lambda $$ λ is a positive parameter, $$f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$ f : Ω × R → R is a continuous function, and $$g\in L^2(0,\infty ;L^2(\Omega ))$$ g ∈ L 2 ( 0 , ∞ ; L 2 ( Ω ) ) . 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.