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

We prove the nonexistence of solutions of the fractional diffusion equation with time-space nonlocal source ut+(−Δ)β2u=(1+|x|)γ∫0t(t−s)α−1|u|p∥ν1q(x)u∥qrds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \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)∈RN×(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x,t) \in \mathbb{R}^{N}\times (0,\infty )$\end{document} with initial data u(x,0)=u0(x)∈Lloc1(RN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u(x,0)=u_{0}(x) \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})$\end{document}, where p,q,r>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p,q,r>1$\end{document}, q(p+r)>q+r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$q(p+r)>q+r$\end{document}, 0<γ≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\gamma \leq 2 $\end{document}, 0<α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\alpha <1$\end{document}, 0<β≤2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0<\beta \leq 2$\end{document}, (−Δ)β2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(-\Delta )^{\frac{\beta }{2}}$\end{document} stands for the fractional Laplacian operator of order β, the weight function ν(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nu (x)$\end{document} is positive and singular at the origin, and ∥⋅∥q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Vert \cdot \Vert _{q}$\end{document} is the norm of Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{q}$\end{document} space.


Introduction
In this paper, we consider the fractional diffusion problem with time-space nonlocal source term of the form u t + (-) subject to the initial data 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 ∈ R N \ {0}, and · q is the norm of the space L q (R N ).
Here we emphasize that equation (1) is a possible model of invasion of a population, and the fractional Laplacian represents the dispersion of the individuals, where u(x, t) repre-sents the density of the species at position x and time t. A nonlocal term is a way to express that the evolution of the species at a point of space depends not only on nearby density but also on the mean value of the total amount of species (see [1]). Equation (1) also suggests the possibility of an interesting physical model in which a superdiffusive medium is coupled to a classically diffusive medium. 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. Such a link may come in the form of a porous material with reactive properties that is partially insulated by contact with a classical diffusive material.
Many researchers have shown a keen interest in the study of differential equations with fractional diffusion in the last few decades. In fact, it has developed into a hot topic of research nowadays. We can find its applications in probability theory, potential theory, fluid dynamics, conformal geometry, mathematical finance, and so on. In addition, fractional diffusion is also important in physics and biology (see [2]).
For the studies on a nonlocal problem, we refer the reader to the paper by Chen et al. [4], who studied some degenerated parabolic inequalities with local and nonlocal nonlinear terms and proved the global nonexistence of nontrivial solutions by the test function method. Furthermore, in [16] the authors considered the homogeneous and inhomogeneous inequalities with singular potential and weighted nonlocal source term and showed the nonexistence of nontrivial global weak solutions for problems (9)-(2) and (10)- (2). The purpose of this paper is establishing the nonexistence of solutions of problem (1)-(2) by the test function method (see [7,12]). This paper is organized as follows. In Sect. 2, we introduce some preliminaries including some definitions and formulation of the main results. The proofs of the main results are presented in Sect. 3.

Preliminaries and main results
In this section, we present some preliminaries and announce the main results. Let us first recall some definitions and properties concerning fractional integrals and derivatives. Let is the Riemann-Liouville fractional integral of order 0 < α < 1 for all f ∈ L q (0, T), 1 ≤ q ≤ ∞.

Proof of main results
Proof of Theorem 2. 9 We assume that u exists globally. In the definition of a weak solution of problem (1)-(2), we take ζ ( Then from (11), using the integration-by-parts formula and the fact that D α 0|t (I α 0|t φ)(x, t) = φ(x, t) in the first term on the left-hand side of (22), we obtain Therefore expression (22) can be written as Now we estimate the right-hand side of inequality (25). First, applying the Hölder inequality to the first term in the right-hand side of (25), we have Applying ε-Young's inequality, we obtain where Similarly, we get the following estimate to the second term: Therefore inequality (25) takes the form Now, to estimate the right-hand side of (26), we consider the change of variables τ = t/T and ξ = x/T 1 β . Then we get Using the fact that (-) Combining (27) and (28) together with u 0 (x) ≥ 0, we arrive at where = θ 1 ( N βθ 2 + sr qβθ 1 -γ βθ 1α -1) + 1. Note that inequality (23) is equivalent to ≤ 0. Thus we consider two cases, < 0 and = 0.
Case < 0. Taking T → ∞ in (29), we get a contradiction which implies that u cannot exist. Case = 0. Passing to the limit as T → ∞ in (29), we obtain On the other hand, repeating the same calculation as before by letting where 1 ≤ R < T is large enough, we do not have R → ∞ as T → ∞. Now we estimate the right-hand side of inequality (25) using again the Hölder and ε-Young inequalities as follows: which, on introducing the change of variables τ = t/T and ξ = x/R -1 T Taking the limit in (30) as T → ∞, we get where = β(1θ 1 (α + 1))β and = β(1θ 1 α)β . Finally, we obtain ∞ 0 R N 1 + |x| γ |u| p ν 1 q (x)u r q dx dt ≤ 0 by taking the limit as R → ∞, which leads to u ≡ 0, and hence u cannot exist. The proof of Theorem 2.9 is completed.
This completes the proof.