Blow-up of solutions to systems of nonlinear wave equations with supercritical sources

Abstract

In this article, we focus on the life span of solutions to the following system of nonlinear wave equations:

in a bounded domain Ω ⊂ ℝ ⁿ with Robin and Dirichlét boundary conditions on u and v, respectively. The nonlinearities f ₁(u, v) and f ₂(u, v) represent strong sources of supercritical order, while g ₁(u _t ) and g ₂(v _t ) represent interior damping. The nonlinear boundary condition on u, namely ∂_ν u + u + g(u _t ) = h(u) on Γ, also features h(u), a boundary source, and g(u _t ), a boundary damping. Under some restrictions on the parameters, we prove that every weak solution to system above blows up in finite time, provided the initial energy is negative.

Keywords:

• blow-up
• nonlinear wave equations
• damping and source terms
• weak solutions
• energy identity

AMS Subject Classifications:

• Primary 35L05, 35L20
• Secondary 58J45