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Critical parameters for reaction-diffusion equations involving space-time fractional derivatives. (English) Zbl 1442.35207

Summary: We will look at reaction-diffusion type equations of the following type, \[ \partial^\beta_tV(t,x)=-(-\Delta )^{\alpha /2} V(t,x)+I^{1-\beta }_t[V(t,x)^{1+\eta }].\] We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when \(0<\eta \leqslant \eta_c\), there is no global solution other than the trivial one while for \(\eta >\eta_c\), non-trivial global solutions do exist. The critical parameter \(\eta_c\) is shown to be \(\frac{1}{\eta^*}\) where \[ \eta^*:=\sup_{a>0}\left\{ \sup_{t\in (0,\,\infty ),x\in \mathbb{R}^d}t^a\int_{\mathbb{R}^d}G(t,\,x-y)V_0(y)\,\text{d}y<\infty \right\} \] and \(G(t,x)\) is the heat kernel of the corresponding unforced operator. \(V_0\) is a non-negative initial function. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the fractional time derivative induces a significant change in the behavior of the solution.

MSC:

35K57 Reaction-diffusion equations
35R11 Fractional partial differential equations
35B44 Blow-up in context of PDEs
35B33 Critical exponents in context of PDEs
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