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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 $$du/dt + A(t) \ni 0$$.

##### MSC:
 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.