Existence of Classical Solutions for a Class of the Khokhlov-Zabolotskaya-Kuznetsov Type Equations

Abstract

In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov--Zabolotskaya--Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of~the~Khokhlov--Zabolotskaya--Kuznetsov type equations for the existence of global classical solutions. We~give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.