Nonautonomous heat equations with generalized Wentzell boundary conditions
The paper deals with the linear problem (C) \[ u_t = A(x,t) u \qquad \text{for} \quad 0 \leq x \leq 1, \quad 0 \leq s < t \leq T, \] \[ u(x,s) = f(x) \quad \text{for} \quad 0 \leq x \leq 1, \] \[ A(j,t) u(j,t) + \beta_j(t) u_x(j,t) + \gamma_j u(j,t) = 0 \quad \text{for} \quad j=0,1, \quad 0 \leq s < t \leq T, \] where \(A(x,t)\) is a linear second order differential operator in \(x\) depending on \(t\). To the operator \(A(x,t)\) the authors associate a family of operators A\((t) (0\leq s < t \leq T)\) in the Banach space \(C[0,1]\) defined as \((\text{A}(t)v)(x) = A(x,t)v(x)\) for \(0 \leq x \leq 1\), where \(v\) is a function of class \(C^2[0,1]\) satisfying for fixed \(t\) the boundary conditions as in (C). The subject of the paper is an abstract Cauchy problem (AC) \[ u'(t) = \text{A}(t) u(t) + F(t) \quad \text{for} \quad 0 \leq s < t \leq T, \qquad u(s) = f, \] where \(F\) is a given function with values in \(C[0,1]\). A family of linear operators \(\{U(t,s)\}_{0 \leq s \leq t \leq T}\) in the Banach space \(C[0,1]\) is called an evolution family for the problem (AC) with \(F=0\) if it satisfies \[ U(t,s) U(s,r) = U(t,s) \quad \text{for} \quad 0 \leq r \leq s \leq t \leq T \] and \[ \frac{\partial}{\partial t} U(t,s) = \text{A}(t) U(t,s) \quad \text{for} \quad 0 \leq s < t \leq T \] together with some regularity conditions (the exact definition given in the paper). The main results of the paper are as follows: (i) there exists an unique evolution family for the problem (AC) with \(F=0\) having some properties listed detailly in the theorem 3.3; (ii) under some regularity assumptions concerning \(f, F\) problem (AC) has an unique solution given by \[ u(t) = U(t,s) f + \int_s^t U(t,\sigma)F(\sigma) d\sigma \] (for exact formulation see Theorem 3.4).
scientific article; zbMATH DE number 1960724