The Blow-Up and Global Existence of Solution to Caputo–Hadamard Fractional Partial Differential Equation with Fractional Laplacian

Abstract

This paper is devoted to studying the blow-up and global existence of the solution to a semilinear time-space fractional diffusion equation, where the time derivative is in the Caputo–Hadamard sense and the spatial derivative is the fractional Laplacian. The mild solution of the considered semilinear equation by a convolution form is obtained, where the fundamental solutions are denoted by Fox H-functions. Then, applying contraction mapping principle, the local existence and uniqueness of the mild solution are shown, and the mild solution is proved to be a weak solution. The blow-up in a finite time and global existence of the solution to this semilinear equation are displayed by a fixed point argument. Finally, the blow-up of solution in a finite time is verified by numerical simulations.

Appendix

In the Appendix, we briefly introduce the definition of Fox H-function, see also (Kilbas et al. 2006; Kilbas and Saigo 2004; Srivastava et al. 1982) in detail.

Let $0\le m \le \nu$ and $0\le n \le \mu$ with $m,n,\mu ,\nu \in \mathbb {Z}$. For $a_{l}, b_{j}\in \mathbb {C}$, and $\alpha _{l}, \beta _{j}\in \mathbb {R}^{+}\ (l=1,\ldots ,\mu ;\ j=1,\ldots ,\nu )$, the Fox H-function $H_{\mu \nu }^{mn}(z)$ is defined via a Mellin–Barnes-type integral as

$$\begin{aligned} H_{\mu \nu }^{mn}(z)&\equiv H_{\mu \nu }^{mn}\bigg (z\bigg | \begin{array}{l} (a_{1},\alpha _{1}),\ldots , (a_{n},\alpha _{n});\,\, (a_{n+1},\alpha _{n+1}), \ldots , (a_{\mu },\alpha _{\mu }) \\ (b_{1},\beta _{1}),\,\ldots , (b_{m},\beta _{m});\, (b_{m+1},\beta _{m+1}), \ldots , (b_{\nu },\beta _{\nu }) \end{array} \bigg ) \\&:=\frac{1}{2\pi i}\int _{\mathcal {C}} \mathcal {H}_{\mu \nu }^{mn}(\tau )z^{-\tau }\mathrm{{d}}\tau , \end{aligned}$$

where

$$\begin{aligned} \mathcal {H}_{\mu \nu }^{mn}(\tau ):=\frac{\prod _{j=1}^{m}\Gamma (b_{j} +\beta _{j}\tau )\prod _{l=1}^{n}\Gamma (1-a_{l}-\alpha _{l}\tau )}{\prod _{l=n+1}^{\mu }\Gamma (a_{l}+\alpha _{l}\tau ) \prod _{j=m+1}^{\nu }\Gamma (1-b_{j}-\beta _{j}\tau )}, \end{aligned}$$

and, the poles

$$\begin{aligned} b_{j\sigma }=-\frac{b_{j}+\sigma }{\beta _{j}} \ \ (j=1, \ldots , m;\,\sigma =0,1,2,\ldots ) \end{aligned}$$

and the poles

$$\begin{aligned} a_{lk}=\frac{1-a_{l}+k}{\alpha _{l}} \ \ (l=1, \ldots , n;\ k=0,1,2,\ldots ) \end{aligned}$$

do not coincide, i.e.,

$$\begin{aligned} \alpha _{l}(b_{j}+\sigma )\ne \beta _{j}(a_{l}-k-1) \ \ (j=1, \ldots , m;\ l=1, \ldots , n; \ \sigma , \,k=0,1,2,\ldots ). \end{aligned}$$

The contour $\mathcal {C}$ is the infinite contour in the complex plane which separates all the poles at $\tau =b_{j\sigma }$ to the left and all the poles at $\tau =a_{lk}$ to the right of $\mathcal {C}$.