# Global existence of weak solutions for strongly damped wave equations with nonlinear boundary conditions and balanced potentials

## TL;DR

This paper proves the global existence of weak solutions for a class of strongly damped wave equations with nonlinear boundary conditions, accommodating supercritical growth and mixed dissipative behaviors under minimal regularity assumptions.

## Contribution

It establishes the existence of weak solutions for complex damped wave equations with nonlinear boundary conditions, allowing supercritical growth and minimal interior regularity.

## Key findings

- Proved global existence of weak solutions.
- Handled supercritical polynomial growth.
- Allowed mixed dissipative and anti-dissipative nonlinearities.

## Abstract

We demonstrate the global existence of weak solutions to a class of semilinear strongly damped wave equations possessing nonlinear hyperbolic dynamic boundary conditions. Our work assumes $(-\Delta_W)^\theta \partial_tu$ with $\theta\in[\frac{1}{2},1)$ and where $\Delta_W$ is the Wentzell-Laplacian. Hence, the associated linear operator admits a compact resolvent. A balance condition is assumed to hold between the nonlinearity defined on the interior of the domain and the nonlinearity on the boundary. This allows for arbitrary (supercritical) polynomial growth on each potential, as well as mixed dissipative/anti-dissipative behavior. Moreover, the nonlinear function defined on the interior of the domain is assumed to be only $C^0$.