Which Kind of Fractional Partial Differential Equations Has Solution with Exponential Asymptotics?

Abstract

It is known that the solutions to the Cauchy problems with integer-order derivative or fractional Caputo derivative defined on $\mathbb {R}^{+}\times \mathbb {R}^{d}$ can behave algebraically asymptotical, and the Cauchy problems with Caputo-Hadamard derivative defined on $\mathbb {R}_{a}^{+}\times \mathbb {R}^{d}$ can behave logarithmically asymptotical, where $\mathbb {R}_{a}^{+}=[a,+\infty ), a>0$. Does there exist a kind of fractional derivatives such that the corresponding Cauchy problems have exponentially asymptotical solution? The answer is positive. We can introduce a new derivative with exponential kernel such that the associate solution is exponentially asymptotic.