On weak solutions of a nonlinear hyperbolic-parabolic partial differential equation
In the cylinder \(Q_\infty= \Omega\times (0, \infty)\) the following mixed problem is considered \[ (K(x, t) u')'+ K_2(x, t) u'+ A(t) u+ F(u)= f,\tag{1} \] \[ u|_{\partial\Omega\times (0, \infty)}= 0,\quad u(x, 0)= u_0(x),\quad u'(x, 0)= u_1(x),\tag{2} \] where \(K_1\geq 0\), \(A(t)\) is an elliptic operator of the second order with smooth coefficients, \(A_t\leq 0\). The existence of weak solutions to (1)--(2) and exponential decay of the energy, as \(t\to \infty\), are proven. Earlier, this problem was considered in the case \(A(t)= -\Delta\) by \textit{A. B. Maciel} [Nonlinear Anal. Theory Methods Appl. 20, No. 6, 745-754 (1993; Zbl 0802.35060)], where the same results were obtained. In the present paper, the author uses a restriction: \(K_1(x, 0)\geq \delta_0> 0\), which implies strong hyperbolicity of (1) in some vicinity of the surface \(t= 0\). This helps to avoid the problem of posing initial conditions at \(t= 0\), which is an essential problem in the theory of degenerating equations and has been considered in papers of \textit{V. N. Vrasov} [Dokl. Akad. Nauk SSSR 224, 273-276 (1975); translation in Sov. Math., Dokl. 16, 1179-1183 (1976; Zbl 0331.35043)], and the reviewer [Sib. Math. J. 18, No. 6, 1003-1006 (1978); translation from Sib. Mat. Zh. 18, 1414-1419 (1977; Zbl 0398.35060)].
scientific article; zbMATH DE number 854783