Abstract
We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal sourceut+(−Δ)β2u=(1+|x|)γ∫0t(t−s)α−1|u|p∥ν1q(x)u∥qrds\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned} u_{t} + (-\\Delta )^{\\frac{\\beta }{2}} u =\\bigl(1+ \\vert x \\vert \\bigr)^{ \\gamma } \\int _{0}^{t} (t-s)^{\\alpha -1} \\vert u \\vert ^{p} \\bigl\\Vert \\nu ^{ \\frac{1}{q}}(x) u \\bigr\\Vert _{q}^{r} \\,ds \\end{aligned}$$ \\end{document} for (x,t) in mathbb{R}^{N}times (0,infty ) with initial data u(x,0)=u_{0}(x) in L^{1}_{mathrm{loc}}(mathbb{R}^{N}), where p,q,r>1, q(p+r)>q+r, 0<gamma leq 2 , 0<alpha <1, 0<beta leq 2, (-Delta )^{frac{beta }{2}} stands for the fractional Laplacian operator of order β, the weight function nu (x) is positive and singular at the origin, and Vert cdot Vert _{q} is the norm of L^{q} space.
Highlights
We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source ut +
Where u = u(x, t) is a real-valued unknown function of (x, t), p, q, r > 1, q(p + r) > q + r, 0 < γ ≤ 2, 0 < α < 1, 0 < β ≤ 2, the weight function ν(x) is positive and singular at the origin, that is, there exist c > 0 and s ≥ 0 such that ν(x) ≥ c|x|–s, x ∈ RN \ {0}, and · q is the norm of the space Lq(RN )
The right-hand side of (1) can be interpreted as the effect of a classical diffusive medium that is nonlinearly linked to a superdiffusive medium
Summary
We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source ut + 1 Introduction In this paper, we consider the fractional diffusion problem with time-space nonlocal source term of the form ut + (– ) β2 u = 1 + |x| γ t(t – s)α–1|u|p ν q1 (x)u rq ds, (x, t) ∈ RN × (0, ∞), (1) Let us recall some known results on fractional diffusion equations.
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