On some class of problems with nonlocal source and boundary flux (Q5954499)

The paper deals with the folowing nonlocal, semilinear, parabolic problem (P): NEWLINE\[NEWLINE\begin{aligned} u_t + Au &= -a(q(u)) \quad \text{in} \quad (0,T) \times \Omega,\\ u(t,.) = 0 \quad \text{on} \quad (0,T) \times \Gamma_0, & \qquad \partial_n u(t,.) = b(q(u)) \quad \text{on} \quad (0,T) \times \Gamma_1,\\ u(0,.) = u_0(.) \quad \text{in} \quad \Omega, & \qquad q(u) \in D \quad \text{on} \quad (0,T). \end{aligned}NEWLINE\]NEWLINE Here \(\Omega \subset \mathbb{R}^N\) is a bounded, Lipschitz domain, \(\Gamma_0\) and \(\Gamma_1\) are two disjoint parts of \(\partial \Omega\) with \(\overline{\Gamma_0} \cup \overline{\Gamma_1} = \partial \Omega\); \(A\) is the linear elliptic operator; \(a,b\) are real-valued functions defined on an interval \(D \subset \mathbb{R}\) and \(q\) is a functional from \(L^2(\Omega)\) into \(\mathbb{R}\). NEWLINENEWLINENEWLINEIn physical applications systems of such kind describe the evolution in time of the temperature \(u(x,t)\) of a thin plate \(\Omega\), when \(\Omega\) is cooled and at this same time some part \(\Gamma_1\) of its border is heated. NEWLINENEWLINENEWLINEAfter formulating problem (P) in variational form the authors prove the existence of its solution for some \(T>0\) (using the Schauder fixed point theorem) and the unicity of this solution. In the sequel of the paper it is proved that the obtained solution may be uniquely extended in time to the so-called maximal solution; some properties of the maximal solution in the case \(N=1\) are investigated.