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 ∂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}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} \\displaystyle \\partial _t^\\beta u+(-\\Delta )^\\alpha u+(-\\Delta )^\\alpha \\partial _t^\\beta u=\\lambda f(x,u) +g(x,t) &{}\ ext{ in } \\Omega \ imes {\\mathbb {R}}^{+},\\\\ u(x,t)=0\\ \\ &{}\ ext{ in } ({\\mathbb {R}}^N{\\setminus }\\Omega )\ imes {\\mathbb {R}}^+,\\\\ u(x,0)=u_0(x)\\ &{}\ ext{ in } \\Omega ,\\\\ \\end{array}\\right. } \\end{aligned}$$\\end{document} where Omega subset {mathbb {R}}^N is a bounded domain with Lipschitz boundary, (-Delta )^{alpha } is the fractional Laplace operator with 0<alpha <1, partial _t^{beta } is the Riemann-Liouville time fractional derivative with 0<beta <1, lambda is a positive parameter, f:Omega times {mathbb {R}}rightarrow {mathbb {R}} is a continuous function, and gin L^2(0,infty ;L^2(Omega )). 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.