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On comparison theorems in nonlinear evolution equation. (English) Zbl 0843.60056

Deng, Zhongqi (ed.) et al., Differential equations and control theory. Proceedings of the international conference, Wuhan, People’s Republic of China, 1994. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 176, 471-475 (1996).
Denote by \(C_b(R^d)\) and \(B(R^d)\) the family of bounded continuous functions and bounded functions, respectively. Let \(T_t\) be a contraction \(C_0\)-semigroup on \(C_b(R^d)\). \(A\) is the infinitesimal generator of \(T_t\). Set: \[ \Psi(x,\lambda) = - b(x)\lambda - c(x)\lambda^2 + \int^\infty_0 (1 - e^{-\lambda u} - \lambda u) n(x,du). \] Here \(b, c \geq 0 \in B(R^d)\), \(n(x,du)\) is a kernel from \(R^d \times R_+\) to \(R_+\) such that \(\int^\infty_0 u \wedge u^2 n(x,du) \in B(R^d)\). Consider the following nonlinear evolution equation: \[ u(t) = T_tf + \int^t_0 T_{t - s} \Psi(u(t))ds.\tag{1} \] Since \(\Psi(x,\lambda)\) is Lipschitz continuous in \(\lambda\), it is well-known that (1) has a unique mild solution \(u\). Moreover, if \(f \in {\mathcal D}(A)\) (the domain of operator \(A\)), then \(u\) also satisfies \[ {du(t)\over dt} = Au(t) + \Psi(u(t)),\qquad u(0) =f. \] The author obtains some comparison results of the solutions of (1) for different \(T_t\) and \(\Psi\). In particular, the comparison results can be applied to the case in which \(A\) is the following second order partial differential operator on \(R^d\), \[ Lf(x) = \sum_{i,j} a_{ij}(x) {\partial^2f(x)\over \partial x_i \partial x_j} + \sum^d_{i = 1} b_i(x) {\partial f(x) \over \partial x_i}, \] where \(a(x) \equiv (a_{ij}(x))\) is positive definite matrix, \(a_{ij}(x) \in C(R^d)\), \(b_i(x) \in B(R^d)\), \(1 \leq i,j \leq d\).
For the entire collection see [Zbl 0833.00030].

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
47H40 Random nonlinear operators
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