Of concern is the nonlinear uniformly parabolic problem \begin{equation*} u_t =\dv(\A\nabla u),\qquad u(0,x)=f(x),\qquad u_t +\beta\pan u+\gamma(x, u)-q\beta \lb u=0, \end{equation*} for $x\in \Omega\subset \R^N$ and $t\ge0$; the last equation holds on the boundary $\p\Omega$. Here $\A=\{a_{ij}(x)\}_{ij}$ is a real, hermitian, uniformly positive definite $N\times N$ matrix; $\beta\in C(\p\Omega)$ with $\beta>0$; $\gamma:\p\Omega\times\R\to \R; \,q\in [0,\infty)$ and $\pan u$ is the conormal derivative of $u$ with respect to $A$: and everything is sufficiently regular. The solution of this wellposed problem depends continuously on the ingredients of the problem, namely, $\A,\,\beta,\,\gamma,\,q,\, f.$ This is shown using semigroup methods in \cite{CFGGR}. Here we prove explicit stability estimates of the solution $u$ with respect to the coefficients $\A,\,\beta,\,\gamma,\,q,$ and the initial condition $f$. Moreover we cover the singular case of a problem with $q=0$ which is approximated by problems with positive $q$.

Stability of Parabolic Problems with nonlinear Wentzell boundary conditions / Coclite, Giuseppe Maria; Goldstein, G. R.; Goldstein, J. A.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 246:6(2009), pp. 2434-2447. [10.1016/j.jde.2008.10.004]

Stability of Parabolic Problems with nonlinear Wentzell boundary conditions

COCLITE, Giuseppe Maria;
2009-01-01

Abstract

Of concern is the nonlinear uniformly parabolic problem \begin{equation*} u_t =\dv(\A\nabla u),\qquad u(0,x)=f(x),\qquad u_t +\beta\pan u+\gamma(x, u)-q\beta \lb u=0, \end{equation*} for $x\in \Omega\subset \R^N$ and $t\ge0$; the last equation holds on the boundary $\p\Omega$. Here $\A=\{a_{ij}(x)\}_{ij}$ is a real, hermitian, uniformly positive definite $N\times N$ matrix; $\beta\in C(\p\Omega)$ with $\beta>0$; $\gamma:\p\Omega\times\R\to \R; \,q\in [0,\infty)$ and $\pan u$ is the conormal derivative of $u$ with respect to $A$: and everything is sufficiently regular. The solution of this wellposed problem depends continuously on the ingredients of the problem, namely, $\A,\,\beta,\,\gamma,\,q,\, f.$ This is shown using semigroup methods in \cite{CFGGR}. Here we prove explicit stability estimates of the solution $u$ with respect to the coefficients $\A,\,\beta,\,\gamma,\,q,$ and the initial condition $f$. Moreover we cover the singular case of a problem with $q=0$ which is approximated by problems with positive $q$.
2009
Stability of Parabolic Problems with nonlinear Wentzell boundary conditions / Coclite, Giuseppe Maria; Goldstein, G. R.; Goldstein, J. A.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 246:6(2009), pp. 2434-2447. [10.1016/j.jde.2008.10.004]
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11589/93859
Citazioni
  • Scopus 30
  • ???jsp.display-item.citation.isi??? 32
social impact