The threshold of effective damping for semilinear wave equations

Abstract

Abstract. In this paper we study the global existence of small data solutions to the Cauchy problemwhere µ ≥ 2. We obtain estimates for the solution and its energy with the same decay rate of the linear problem. We extend our results to a model with polynomial speed of propagation, and to a model with an exponential speed of propagation and a constant damping ν ut.

A shift in the Strauss exponent for semilinear wave equations with a not effective damping

“…We prove the following results: Being the 1-dimensional existence result already proved in [3], we prove the existence result for space dimension n = 2 and n = 3. …”

“…Let µ ∈ (0, 2) in (3). We may expect that the critical exponent p µ (n) is not larger thanp µ (n) due to the fact that the model in (67) with m = (µ−2)µ has an additional negative mass term with respect to the model in (3). Moreover, we know that the critical exponent has to be not smaller than p ∞ (n − (1 − µ) + ).…”

“…Indeed, if n = 1 our considerations are restricted to µ ∈ (0, 5/3), since we already know that the critical exponent is 3 for µ ≥ 5/3. On the other hand, if n ≥ 3 and µ ∈ (2, n + 2) in (3), then we may expect that the critical exponent p µ (n) is not smaller thanp µ (n), due to the fact that the model in (67) with m = (µ−2)µ has an additional positive mass term with respect to the model in (3). Moreover, we know that the critical exponent may not be smaller than p ∞ (n).…”

“…• According to Theorems 2 and 3 in [3] global (in time) existence of energy solutions for small data if p > 1 + 2/n and -if n = 1 and µ ≥ 5/3, -if n = 2 and µ ≥ 3, -for any n ≥ 3 if µ ≥ n + 2.…”

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation

“…We recall only results for . It was recently shown in , that when if , if and if , then is the critical exponent as in the general case with effective dissipation . Note that the assumptions for μ 1 imply that the damping term is not non‐effective according to the classification of Wirth .…”