Remarks on a nonlocal problem involving the Dirichlet energy. (English) Zbl 1117.35034

The authors study the following nonlinear parabolic problem of nonlocal type depending on the Dirichlet integral: find \(u=u\left( x,t\right) \) solution to \[ u_t-a\left( \int_\Omega \left| \nabla u\right| ^2\,dx\right) \Delta u=f\;\;in\;\Omega \times \mathbb R^+, \]
\[ u\left( x,0\right) =u_0\left( x\right) \;in\;\Omega ,\;\;u=0\;on\;\partial \Omega \times R^{+}, \] where \(\Omega \) is a bounded, smooth open subset of \(\mathbb R^n\), \(n\geq 1\), \(f\in L^2( \Omega )\), \(u_0\in H_0^1(\Omega)\) and \(a=a(s)\) is a continuous function such that \(0<m\leq a\left( s\right) \leq M\). After proving the uniqueness and existence of the solution, the authors study the corresponding stationary problem and finally give some results concerning the asymptotic behaviour of the solution.


35K55 Nonlinear parabolic equations
35K20 Initial-boundary value problems for second-order parabolic equations
35J20 Variational methods for second-order elliptic equations
35R10 Partial functional-differential equations
35B40 Asymptotic behavior of solutions to PDEs
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