Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping
Abstract
In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both |u|p and |ut|p. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range 1 < p ≤ pStr(n) for the power of the nonlinear term |u|p. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case p=pStr(n), we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity |ut|p we prove the blow-up of weak solutions in the sub-Glassey range 1 < p ≤ (n+1)/(n−1) by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.
Related
- Oversimplified dislocation dynamics from semilinear homogenization. Heat and damped wave equations
- An exact traveling waves solution for a special class of semilinear equations
- Interior regularity of mixed local-nonlocal parabolic semilinear equations
- Liouville Theorems for Semilinear Parabolic Equations Involving the Nonlinear Gradient Term
- Weyl Pseudo Almost Periodic Type Solutions to Semilinear Stochastic Evolution Equations Driven by Fractional Brownian Motion
BibTeX
@misc{nascimento2026strauss,
title = {{Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping}},
author = {Wanderley Nunes do Nascimento and Alessandro Palmieri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.14311},
eprint = {2608.14311},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.14311},
}
RIS
TY - UNPB
TI - Strauss and Glassey exponents for semilinear wave equations with a non-effective and not scattering producing damping
AU - Wanderley Nunes do Nascimento
AU - Alessandro Palmieri
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2608.14311
UR - https://arxiv.org/abs/2608.14311
AB - In this paper, we study the Cauchy problems for a semilinear damped wave equation with a time-dependent coefficient for the damping term belonging to the class of non-effective damping terms with critical decay rates involving iterated logarithmic factors. As nonlinearities we consider both |u|^p and |u_t|^p. Assuming nonnegative and compactly supported initial data, we establish the blow-up for weak solutions in the sub-Strauss range 1 < p ≤ p_Str(n) for the power of the nonlinear term |u|^p. The proof in the sub-critical case relies on an iteration frame for the space average of the solution, obtained by employing a time-dependent multiplier related to the coefficient of the damping term. On the other hand, in the limit case p=p_Str(n), we work with solutions of the homogeneous equation with separated variables, and we investigate the properties of a fundamental system of solutions for the corresponding time-dependent ODE, in order to derive an iteration frame for a suitable weighted space average of the solution. Finally, for the derivative type nonlinearity |u_t|^p we prove the blow-up of weak solutions in the sub-Glassey range 1 < p ≤ (n+1)/(n−1) by using a comparison argument for a suitable time-dependent function associated with the corresponding local in time solution.
ER -