<p>In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$|u|^{1+\frac{2}{n}}\mu (|u|)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mfrac> <mn>2</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">|</mo> <mi>u</mi> <mo stretchy="false">|</mo> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> is a modulus of continuity. In recent papers by Ebert–Girardi–Reissig (Math. Ann. 378, 1311–1326, 2020) and Girardi (Nonlinear Differ. Equ. Appl. 32, 3, 2025), the authors obtained a sharp critical condition on <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> in low space dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n=1,2,3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, which determines the threshold between global (in time) existence of small data solutions and blow-up of solutions in finite time. Our new results are to prove that this condition remains valid in dimension <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation>, together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$t \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> is identified by the Gauss kernel. Finally, a sharp lower lifespan estimate for blow-up solutions is also derived in the case when blow-up occurs.</p>

On the Asymptotic Profile of Solutions to Semilinear Damped Wave Equations with Critical Nonlinearities

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发表日期：

2026年6月30日

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Trung Loc Tang,
Dinh Van Duong

摘要

In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+\frac{2}{n}}\mu (|u|)$ ${| u |}^{1 + \frac{2}{n}} μ (| u |)$ , where $\mu $ $μ$ is a modulus of continuity. In recent papers by Ebert–Girardi–Reissig (Math. Ann. 378, 1311–1326, 2020) and Girardi (Nonlinear Differ. Equ. Appl. 32, 3, 2025), the authors obtained a sharp critical condition on $\mu $ $μ$ in low space dimensions $n=1,2,3$ $n = 1, 2, 3$ , which determines the threshold between global (in time) existence of small data solutions and blow-up of solutions in finite time. Our new results are to prove that this condition remains valid in dimension $n=4$ $n = 4$ , together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at $t \rightarrow \infty $ $t \to \infty$ is identified by the Gauss kernel. Finally, a sharp lower lifespan estimate for blow-up solutions is also derived in the case when blow-up occurs.