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

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