Global Solvability of a Mixed Problem for a Nonlinear Hyperbolic-Parabolic Equation in Noncylindrical Domains

Jorge Ferreira and Nickolai A. Lar'kin

Abstract: In this paper we study the global existence and uniqueness of regular solutions to the mixed problem for the nonlinear hyperbolic-parabolic equation $$ \eqalign{&{}K_1(x,t)\,\utt + K_2 (x,t)\,\ut - \De u + f_1 (t)\,|u|^\rho\,u=f(x,t)\hbox{in}\hqm,\cr \noalign{\vskip1.5mm} &{}u=0\hbox{at}\hsigm_t,\cr \noalign{\vskip1.5mm} &{}u (x,0)=u_0 (x), \ut (x,0)=\uu (x), x\in\omm_0,\cr} $$ where $\hqm$ is a noncylindrical domain of $\crr^{n+1}$ with the lateral boundary $\hsigm_t$ and $K_1$, $K_2$, $f_1$ are functions which satisfy some appropriate conditions.

Keywords: Noncylindrical domains; regular solutions; nonlinear hyperbolic-parabolic equation.

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