Nonexistence of global solutions of fractional diffusion equation with time-space nonlocal source

Abstract

Abstract We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source u t + ( − Δ ) β 2 u = ( 1 + | x | ) γ ∫ 0 t ( t − s ) α − 1 | u | p ∥ ν 1 q ( x ) u ∥ q r d s $$\begin{aligned} u_{t} + (-\Delta )^{\frac{\beta }{2}} u =\bigl(1+ \vert x \vert \bigr)^{ \gamma } \int _{0}^{t} (t-s)^{\alpha -1} \vert u \vert ^{p} \bigl\Vert \nu ^{ \frac{1}{q}}(x) u \bigr\Vert _{q}^{r} \,ds \end{aligned}$$ for ( x , t ) ∈ R N × ( 0 , ∞ ) $(x,t) \in \mathbb{R}^{N}\times (0,\infty )$ with initial data u ( x , 0 ) = u 0 ( x ) ∈ L loc 1 ( R N ) $u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$ , where p , q , r > 1 $p,q,r>1$ , q ( p + r ) > q + r $q(p+r)>q+r$ , 0 < γ ≤ 2 $0<\gamma \leq 2 $ , 0 < α < 1 $0<\alpha <1$ , 0 < β ≤ 2 $0<\beta \leq 2$ , ( − Δ ) β 2 $(-\Delta )^{\frac{\beta }{2}}$ stands for the fractional Laplacian operator of order β, the weight function ν ( x ) $\nu (x)$ is positive and singular at the origin, and ∥ ⋅ ∥ q $\Vert \cdot \Vert _{q}$ is the norm of L q $L^{q}$ space.

Keywords

• Nonlocal source
• Test function
• Nonexistence of global solution