We study existence and global behavior of solutions to a wave equation with exponential growth source term and locally distributed nonlinear dissipation posed in a bounded domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation>. The results of this paper are the local wellposedness in the energy space, uniform decay of the energy as <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> in the defocusing case (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$+f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>+</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>), and the dichotomy into global existence/uniform decay and blow-up (for <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$a=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) in the focusing case (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$-f(u)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>) for those solutions with energy less than d of the ground state, where d is the level of the Mountain Pass Theorem. We give a proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof is that the decay rate is independent of the nonlinearity.

Uniform Decay Rates and Blow-up for the Wave Equation with Nonlinear Dissipation and Exponential Source

期刊论文
发表日期：

2026年3月12日

查看全文

Marcelo Moreira Cavalcanti,
José Guilherme Simion Antunes

摘要

We study existence and global behavior of solutions to a wave equation with exponential growth source term and locally distributed nonlinear dissipation posed in a bounded domain $\Omega \subset \mathbb {R}^2$ $Ω \subset R^{2}$ . The results of this paper are the local wellposedness in the energy space, uniform decay of the energy as $t\rightarrow \infty $ $t \to \infty$ in the defocusing case ( $+f(u)$ $+ f (u)$ ), and the dichotomy into global existence/uniform decay and blow-up (for $a=1$ $a = 1$ ) in the focusing case ( $-f(u)$ $- f (u)$ ) for those solutions with energy less than d of the ground state, where d is the level of the Mountain Pass Theorem. We give a proof based on the truncation of the original problem and passage to the limit in order to obtain in one shot, the energy identity as well as the Observability Inequality, which are the essential ingredients to obtain uniform decay rates of the energy. One advantage of our proof is that the decay rate is independent of the nonlinearity.