Life-span of solutions to semilinear wave equation with time-dependent critical damping for specially localized initial data
By Masahiro Ikeda, M. Sobajima. Mathematische Annalen, 2017
96 citations recorded by the source corpus.
This paper is concerned with the blowup phenomena for initial value problem of semilinear wave equation with critical time-dependent damping term DW∂t2u(x,t)-Δu(x,t)+μ1+t∂tu(x,t)=|u(x,t)|p,(x,t)∈RN×(0,T),u(x,0)=εf(x),x∈RN,∂tu(x,0)=εg(x),x∈RN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^2 u(x,t) -\Delta u(x,t) + \dfrac{\mu }{1+t}\partial _t u(x,t)=|u(x,t)|^p, &{} (x,t)\in \mathbb {R}^N \times (0,T),\\ u(x,0)=\varepsilon f(x),&{} x\in \mathbb {R}^N,\\ \partial _t u(x,0)=\varepsilon g(x),&{} x\in \mathbb {R}^N, \end{array}\right. } \end{aligned}$$\end{document}where N∈N,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\in \mathbb {N},$$\end{document}μ∈[0,N2+N+2N+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu \in [0,\frac{N^2+N+2}{N+2})$$\end{document} and ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is a parameter describing the smallness of initial data. Given data f, g are compactly supported in a small area inside of the unit ball. The result is the upper bound of lifespan of solution u with respects to the small parameter ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} when pF(N)≤p≤p0(N+μ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_F(N)\le p\le p_0(N+\mu ),$$\end{document} where pF(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_F(N)$$\end{document} denotes the Fujita exponent for the nonlinear heat equations and p0(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0(n)$$\end{document} denotes the Strauss exponent for nonlinear wave equation (DW) in n-dimension with μ=0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =0.$$\end{document} Consequently, by connecting the result of D’Abbicco–Lucente–Reissig (J Differ Equ 259:5040–5073, 2015), our result clarifies the threshold exponent p0(N+μ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_0(N+\mu )$$\end{document} for dividing blowup phenomena and global existence of small solutions when N=3.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=3.$$\end{document} The crucial idea is to construct suitable test functions satisfying the conjugate linear equation ∂t2Φ-ΔΦ-∂t(μ1+tΦ)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial _t^2\Phi -\Delta \Phi -\partial _t(\frac{\mu }{1+t}\Phi )=0$$\end{document} of (DW) including the Gauss hypergeometric functions; note that the construction of test functions is different from Zhou–Han (Commun Partial Differ Equ 39:439–451, 2014).
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/3b5259e0bd50056b1b723cacff52ffd3dbc0ae70
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Citations in the corpus
- A note on the blowup of scale invariant damping wave equation with sub-Strauss exponent (2017, background): “As the blowup result of Strauss critical exponent p = pS(n + µ) has been given by Ikeda and Sobajima [4], we note that concerning the determining of critical exponent of scale invariant damping wave equation, the situation of p > ps(n+ µ) for 0 µ ≤ µ∗ needs to be further investigated.”
- Life-span of semilinear wave equations with scale-invariant damping: Critical Strauss exponent case (2017, unknown): “Ikeda-Sobajima [6] obtained similar result.”