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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.

MSC:
 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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References:
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