Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system

2024/04/05 by Benabbas, Imen, Said-Houari, Belkacem
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.03920

Abstract

In this work, we investigate the global existence and asymptotic behavior of a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). First, we prove that the solution exists globally in time, provided that the lower-order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This is done using a continuity argument together with some interpolation inequalities. Second, we prove an exponential decay of the solution under the same smallness assumptions on the initial data.

Related

• The Westervelt--Pennes--Cattaneo model: local well-posedness and singular limit for vanishing relaxation time
• Local well-posedness of a coupled Jordan-Moore-Gibson-Thompson-Pennes model of nonlinear ultrasonic heating
• Local existence results for the Westervelt equation with nonlinear damping and Neumann as well as absorbing boundary conditions
• L_p-L_q-theory for a quasilinear non-isothermal Westervelt equation
• Well-posedness and long-time behavior for the Westervelt equation with\n absorbing boundary conditions of order zero