turkmath.org

In this talk, we are going to study the following Cauchy problem for the plate equation with mass term: $$u_{tt} + \Delta^2u + u = 0\ \text{if} \ (t,x)\in (0,\infty)\times \mathbb{R}^n$$ $$u(0,x)=u_0(x)\ \ u_t(0,x)=u_1(x)\ \ \text{if} \ x\in \mathbb{R}^n.$$ Our goal is to derive $L^p-L^q$ estimates for the solutions to problem \eqref{eq_1} in the full range $1 \leq p \leq q \leq \infty$. After that, we study the associated semilinear model with power non-linearity $|u|^{\alpha}$ with $\alpha>1$, where we apply the derived $L^p-L^q$ estimates in the analysis of local (in time) existence of solution and for the global (in time) small data solutions. The results of this talk are based on collaboration with} Alexandre A. Junior (University of S\~ao Paulo, Ribeir\~ao Preto, Brazil), Marcelo R. Ebert (University of S\~ao Paulo, Ribeir\~ao Preto, Brazil), Antonio Lagioia (University of Bari, Italy).

References

[1] M. D'Abbicco, M.R. Ebert, $L^p-L^q$ estimates for a parameter-dependent multiplier with oscillatory and diffusive components. J. Math. Anal. Appl. {504} (2021).

[2] M. Ikeda, T. Inui, A remark on non-existence results for the semi-linear damped Klein-Gordon equations. RIMS Kôkyûroku Bessatsu B56, (2016), 11--30.

[3] B. Marshall, W. Strauss, S. Wainger, $L^p-L^q$ estimates for the Klein-Gordon equation. J. Math. Pures Appl. 59. (1980) 417--440.