A note on the Fujita exponent in fractional heat equation involving the Hardy potential. (English) Zbl 1487.35049
Summary: In this work, we are interested on the study of the Fujita exponent and the meaning of the blow-up for the fractional Cauchy problem with the Hardy potential, namely,
\[
u_t + ( - \Delta )^su = \lambda \frac{u}{|x|^{2s}} + {u^p} \text{ in } \mathbb{R}^N, \quad u(x,0) = {u_0}(x) \text{ in } \mathbb{R}^N,
\]
where \(N>2s\), \(0<s<1\), \((- \Delta )^s\) is the fractional Laplacian of order \(2s\), \( \lambda>0 \), \(u_0 \geq 0\), and \(1 < p < p_+(s,\lambda)\), where \(p_+( \lambda,s) \) is the critical existence power to be given subsequently.
MSC:
| 35B33 | Critical exponents in context of PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K67 | Singular parabolic equations |
| 35R11 | Fractional partial differential equations |
Keywords:
Fujita exponent; fractional Cauchy heat equation with Hardy potential; blow-up; global solutionReferences:
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