Stability properties of Radon measure-valued solutions for a class of nonlinear parabolic equations under Neumann boundary conditions. (English) Zbl 1509.35143
Summary: In this paper, we address the existence, uniqueness, decay estimates, and the large-time behavior of the Radon measure-valued solutions for a class of nonlinear strongly degenerate parabolic equations involving a source term under Neumann boundary conditions with bounded Radon measure as initial data.
\[
\begin{cases}
u_t = \Delta\psi(u)+h(t)f(x, t) &\quad\text{in } \Omega\times(0, T), \\
\frac{\partial\psi(u)}{\partial\eta} = g(u) &\quad\text{on } \partial\Omega\times(0, T), \\
u(x, 0) = u_0(x) &\quad\text{in } \Omega,
\end{cases}
\]
where \(T >0\), \(\Omega\subset \mathbb{R}^N\) \((N\ge 2)\) is an open bounded domain with smooth boundary \(\partial\Omega\), \(\eta\) is an outward normal vector on \(\partial\Omega \). The initial value data \(u_0\) is a nonnegative bounded Radon measure on \(\Omega \), the function \(f\) is a solution of the linear inhomogeneous heat equation under Neumann boundary conditions with measure data, and the functions \(\psi\), \(g\) and \(h\) satisfy the suitable assumptions.
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K59 | Quasilinear parabolic equations |
| 35K61 | Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 28A33 | Spaces of measures, convergence of measures |
| 28A50 | Integration and disintegration of measures |
Keywords:
Radon measure-valued solutions; nonlinear strongly degenerate parabolic equations; linear inhomogeneous heat equation; Neumann boundary conditions; large-time behaviorReferences:
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