| Full text | |
| Author(s): |
Total Authors: 2
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| Affiliation: | [1] Univ Bari, Dept Math, Via E Orabona 4, I-70125 Bari - Italy
[2] Univ Sao Paulo, Dept Comp & Matemat, BR-14040901 Ribeirao Preto, SP - Brazil
Total Affiliations: 2
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| Document type: | Journal article |
| Source: | NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS; v. 215, FEB 2022. |
| Web of Science Citations: | 0 |
| Abstract | |
In this paper, we find the critical exponent for the existence of global small data solutions to: [u(tt) + (-Delta)(sigma)u + (-Delta)(theta/2) u(t) = f(u, u(t)), t >= 0, x epsilon R-n, (u, u(t))(0, x) = (0, u(1)(x)), in the case of so-called non-effective damping, theta epsilon (sigma, 2 sigma Sigma], where sigma not equal 1 and f = vertical bar u vertical bar(alpha) or f = vertical bar u(t)vertical bar(alpha), in low space dimension. By critical exponent we mean that global small data solution exists for supercritical powers alpha > (alpha) over tilde and do not exist, in general, for subcritical powers 1 < alpha < (alpha) over tilde. Assuming initial data to be small in L-1 or in some other Lp space, p epsilon (1, 2), in addition to the energy space, the critical exponent only depends on the ratio n/(Sigma p). We also prove the global existence of small data solutions in high space dimension for alpha > (alpha) over bar, but we leave open to determine if a counterpart nonexistence result for alpha < <(alpha)over tilde> holds or not. (C)2021 Published by Elsevier Ltd. (AU) | |
| FAPESP's process: | 20/08276-9 - The stationary phase method and applications to evolution partial differential equations |
| Grantee: | Marcelo Rempel Ebert |
| Support Opportunities: | Regular Research Grants |