Critical regularity of nonlinearities in semilinear classical damped wave equations
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Mathematische Annalen
Critical regularity of nonlinearities in semilinear classical damped wave equations M. R. Ebert1 · G. Girardi2 · M. Reissig3 Received: 17 April 2019 / Revised: 30 September 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this paper we consider the Cauchy problem for the semilinear damped wave equation u tt − Δu + u t = h(u), u(0, x) = φ(x), u t (0, x) = ψ(x), 2
where h(s) = |s|1+ n μ(|s|). Here n is the space dimension and μ is a modulus of continuity. Our goal is to obtain sharp conditions on μ to obtain a threshold between global (in time) existence of small data solutions (stability of the zero solution) and blow-up behavior even of small data solutions. Mathematics Subject Classification 35L05 · 35L71 · 35B44
1 Introduction In [12], the authors proved the global existence of small data energy solutions for the semilinear damped wave equation
Communicated by Y. Giga.
B
M. Reissig [email protected] M. R. Ebert [email protected] G. Girardi [email protected]
1
Departamento de Computação e Matemática, Universidade de São Paulo (USP), FFCLRP, Av. dos Bandeirantes, 3900, Ribeirão Prêto, SP CEP 14040-901, Brazil
2
Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via Orabona 4, Bari, Italy
3
Faculty for Mathematics and Computer Science, Technical University Bergakademie Freiberg, Prüferstr. 9, 09596 Freiberg, Germany
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u tt − Δu + u t = |u| p , u(0, x) = φ(x), u t (0, x) = ψ(x),
(1)
in the supercritical range p > 1+ n2 , by assuming compactly supported small data from the energy space. Under additional regularity the compact support assumption on the data can be removed. By assuming data in Sobolev spaces with additional regularity L 1 (Rn ), a global (in time) existence result was proved in space dimensions n = 1, 2 in [5], by using energy methods, and in space dimension n ≤ 5 in [9], by using L r − L q estimates, 1 ≤ r ≤ q ≤ ∞. Nonexistence of general global (in time) small data solutions is proved in [12] for 1 < p < 1 + n2 and in [13] for p = 1 + n2 . The exponent 1 + n2 is well known as Fujita exponent and it is the critical power for the following semilinear parabolic Cauchy problem (see [2]): vt − v = v p , v(0, x) = v0 (x) ≥ 0.
(2)
If one removes the assumption that the initial data are in L 1 (Rn ) and we only assume that they are in the energy space, then the critical exponent is modified to 1 + n4 or to m n 1 + 2m n under additional regularity L (R ), with m ∈ [1, 2]. For the classical damped wave equation, this phenomenon has been investigated in [4]. The diffusion phenomenon between linear heat and linear classical damped wave models (see [3,7,9,10]) explains the parabolic character of classical damped wave models with power nonlinearities from the point of decay estimates of solutions. In the mathematical literature (see for instance [1]) the situation is in general described as follows: We have a semilinear Cauchy problem L(∂t , ∂x , t, x)u = |u| p , u(0, x) = φ(x), u t (
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