Non-existence of global solutions for certain class of fractional evolution equations

ABSTRACT

This paper investigates the problem of non-existence of global solutions for a general spatial–temporal fractional evolution equation: $\begin{array}{rl}& \rho (x){}^c{D}_{0/t}^{\alpha }u+(-\mathrm{\Delta }{)}^{\beta /2}u-\mathrm{\Delta }{u}^m+a\cdot \mathrm{\nabla }{u}^q=h(x,t)|u{|}^p,\\ & (x,t)\in R^N\times (0,+\mathrm{\infty }).\end{array}$Using the test function method, necessary and sufficient conditions are established. The results are extended to a class of an incommensurate fractional spatial–temporal system.

KEYWORDS:

• Fujita critical exponent
• nonlinear evolution equations
• fractional derivatives
• incommensurate fractional system

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 34A08
• 35B44
• 35R11
• 37L05

Acknowledgments

The authors thank Professor Mokhtar Kirane for his suggestions and for his helpful advice and for the stimulating discussions during the preparation of the early draft of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).