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Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. (English) Zbl 0984.35066
The authors study some class of nonlinear nonlocal elliptic and parabolic problems. Precisely, let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with smooth boundary \(\partial Q\), and let \(\Gamma_0\) be a subset of \(\partial \Omega\) having a positive superficial measure. Set \(V=\{v\in H^1 (\Omega)\mid v=0\) on \(\Gamma_0\}\), and consider \(m\) functionals \(q_1,\dots, q_m:V\to \mathbb{R}\), where each \(q_i\) is a positive homogeneous function of degree \(\alpha_i (\in \mathbb{R})\). For a positive function \(a:\mathbb{R}^m \to\mathbb{R}\), consider the problem: \[ -a\bigl(q_1(u), \dots,q_m(u) \bigr)Au= f\text{ in }\Omega, \tag{1} \]
\[ u=0\text{ on }\Gamma_0,\;\partial_\nu u=0\text{ on }\partial\Omega \setminus\Gamma_0, \tag{2} \] where \(f\) is an element of \(V'\), the dual of \(V\), \(A\) is a linear elliptic operator in divergence form, and \(\partial_\nu u\) denotes the conormal derivative of \(u\). It is assumed that the bilinear form canonically associated to \(A\) is coercive on \(V\). A typical example of \(A\) is the Laplacian \(\Delta\). It is shown that this problem has as many solutions as the system of equations in \(\mu=(\mu_1, \dots,\mu_m) \in\mathbb{R}^m\): \[ a^{\alpha_i} (\mu)\mu_i=q_i (\varphi),\;i=1, \dots,m, \] where \(\varphi\in V\) is the unique solution of the problem: \[ -A\varphi= f\text{ in }\Omega, \quad \varphi=0 \text{ on }\Gamma_0,\;\partial_\nu \varphi=0 \text{ on }\partial \Omega\setminus \Gamma_0. \] When \(\Gamma_0= \partial\Omega\) and \(V=H^1_0 (\Omega)\), also the parabolic problem associated to problem (1)-(2) is solved and in particular it is shown that the solutions of the parabolic problem can quench in the sense that they can vanish identically at some finite time.

MSC:
35J65 Nonlinear boundary value problems for linear elliptic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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