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On a parabolic strongly nonlinear problem on manifolds. (English) Zbl 1183.58025
The authors prove existence, uniqueness and asymptotic behavior of weak solutions to the system
\[ \begin{cases} -\sum_{i=1}^n {\partial\over{\partial x_i}} \left( \left|{{\partial u}\over{\partial x_i}}\right|^{p-2} {{\partial u}\over{\partial x_i}}\right)=0 & \text{in}\;Q=\Omega\times(0,T),\\ u=0 & \text{on}\;\Sigma_0=\Gamma_0\times(0,T),\\ {{\partial u}\over{\partial t}}+\sum_{i=1}^n \left|{{\partial u}\over{\partial x_i}}\right|^{p-2} {{\partial u}\over{\partial x_i}}\nu_i+|u|^pu=f & \text{on}\;\Sigma_1=\Gamma_1\times(0,T),\\ u(x,0)=u_0(x) & \text{on}\;\Gamma, \end{cases} \] where \(p>2,\) \(\Gamma_0\cup\Gamma_1=\partial\Omega\) of a domain \(\Omega\subset\mathbb R^n\), \(\overline{\Gamma_1}\cap\overline{\Gamma_1}=\emptyset,\) and \(\Gamma_0,\) \(\Gamma_1\) have positive Lebesgue measures. \(u_0\) and \(f\) are given functions defined on \(\Gamma\) and \(\Sigma_1\), respectively.
MSC:
58J35 Heat and other parabolic equation methods for PDEs on manifolds
35L85 Unilateral problems for linear hyperbolic equations and variational inequalities with linear hyperbolic operators
35L05 Wave equation
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
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