Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation

Mohamed Jleli

doi:10.1515/math-2020-0089

Abstract

Abstract In this paper, we consider the Cauchy problem for a second-order nonlinear equation with mixed fractional derivatives related to the fractional Khokhlov-Zabolotskaya equation. We prove the nonexistence of a classical local in time solution. The obtained instantaneous blow-up result is proved via the nonlinear capacity method.

Highlights

The study of the blow-up phenomena for many initial-boundary value problems that arise in particular in the theory of ion sound waves in plasma [1], waves in semiconductors [2] and electric oscillations [3] was detailed in many studies with the aim to obtain global or local unsolvability results by using the method of test functions
Let u0 and u1 be two functions defined on 3 such that u0 ∈ L2( 3) and u0u1 ∈ L1( 3)
– If u0 ≡ 0 a.e. and u1 ≢ 0, the instantaneous blow-up occurs for any solution to (1)–(2)

Summary

Introduction

The study of the blow-up phenomena for many initial-boundary value problems that arise in particular in the theory of ion sound waves in plasma [1], waves in semiconductors [2] and electric oscillations [3] was detailed in many studies with the aim to obtain global or local unsolvability results by using the method of test functions.The Khokhlov-Zabolotskaya equation arose in the theory of nonlinear underwater acoustics. In [5], the Cauchy problem for the aforementioned second-order nonlinear equation with mixed derivatives with the initial data u(0, w) = u0(w) and ut(0, w) = u1(w), for w ∈ 3, was studied. Based on the nonlinear capacity method developed in [6], in this paper, we obtain an instantaneous blow-up result for the fractional version of the Khokhlov-Zabolotskaya equation. * Corresponding author: Mohamed Jleli, Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia, e-mail: jleli@ksu.edu.sa

Results

Conclusion