On the asymptotic profile of solutions to semilinear damped wave equations with critical nonlinearities
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| Format: | Preprint |
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2025
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| author | Tang, Trung Loc Van Duong, Dinh |
| author_facet | Tang, Trung Loc Van Duong, Dinh |
| contents | In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} μ(|u|)$, where $μ$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020)) and Girardi (Nonlinear Differ. Equ. Appl. 32 (2025)), the authors obtained a sharp critical condition on $μ$ in low space dimensions $n=1,2,3$, 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 $n=4$, together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at $t \to \infty$ is identified by the Gauss kernel. Finally, a sharp lifespan estimate for local solutions is also derived in the case when blow-up occurs. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_19835 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the asymptotic profile of solutions to semilinear damped wave equations with critical nonlinearities Tang, Trung Loc Van Duong, Dinh Analysis of PDEs 35A01, 35B33, 35B44, 35K91 In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} μ(|u|)$, where $μ$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020)) and Girardi (Nonlinear Differ. Equ. Appl. 32 (2025)), the authors obtained a sharp critical condition on $μ$ in low space dimensions $n=1,2,3$, 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 $n=4$, together with the asymptotic profiles of global solutions. From this, we see that the behavior of the solution at $t \to \infty$ is identified by the Gauss kernel. Finally, a sharp lifespan estimate for local solutions is also derived in the case when blow-up occurs. |
| title | On the asymptotic profile of solutions to semilinear damped wave equations with critical nonlinearities |
| topic | Analysis of PDEs 35A01, 35B33, 35B44, 35K91 |
| url | https://arxiv.org/abs/2509.19835 |