zbMATH — the first resource for mathematics

Further investigation of the relation of the operator \(\partial/\partial\sigma+ \partial/\partial\tau\) to evolution governed by accretive operators. (English) Zbl 0811.35172
From the introduction: Crandall and Evans proved existence of mild solution \(u(t)\) to the nonlinear Cauchy problem \[ {du(t) \over dt} + Au(t) \ni f(t) \quad (0 \leq t \leq T), \quad u(0) = x_ 0 \in \overline {\text{Dom} (A)}. \tag{1} \] Here \(A\) is a multivalued accretive operator on a Banach space \(X\) and \(f \in L^ 1 (0,T;X)\). By a mild solution, we mean \(u\) is a limit of difference approximations to (1). We obtain a mild solution to the Cauchy problem \[ {du(t) \over dt} + Au(t) \ni F \bigl( t,u(t) \bigr) \quad (0 \leq t \leq T), \quad u(0) = x_ 0 \in \overline {\text{Dom} (A)} \tag{2} \] with \(A\), as in (1), accretive on \(X\). (2) includes the case \(F = f\) of (1) and other cases that are of greater complexity. Informally, given functions \(\omega\) and \(h\), consider the partial differential equation on \((0,S] \times (0,T]\) \[ {\partial v \over \partial \sigma} (\sigma, \tau) + {\partial v \over \partial \tau} (\sigma, \tau) = h(\sigma, \tau) \tag{3} \] with boundary condition \(v(\sigma, \tau) = \omega (\tau - \sigma)\), when \(\sigma \tau = 0\). For each positive integer \(m\) and \(n\) let \(v_{m,n}\) be the difference approximation to \(v\) obtained when \(\partial \sigma = 1/m\) and \(\partial \tau = 1/n\) in (3). Likewise, in either (1) or (2), let \(u_ n\) denote the difference approximation to \(u\) when \(dt\) is replaced by \(1/n\). Then for the right choice of \(\omega\) and \(h\) – which will depend on \(f\) in (1) and on \(F\) in (2), an estimate of \(u_ m - u_ n\) in terms of \(v_{m,n}\) can be obtained which forces \(\| u_ m - u_ n \|\) to zero as \(v_{m,n}\) converges to \(v\). In this way \(\{u_ n\}_ n\) is shown to be Cauchy and the mild solution \(u = \lim_ n u_ n\) is shown to exist. The emphasis in this work is directed towards methods and estimates tied to convergence of \(v_{m,n}\) to \(v\). Another paper based on the theory presented here will be devoted to the study of the nonautonomous Cauchy equation \(du/dt + A(t) \ni 0\).

35R70 PDEs with multivalued right-hand sides
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
47J05 Equations involving nonlinear operators (general)
47H06 Nonlinear accretive operators, dissipative operators, etc.