Abstract
In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients.
Highlights
Let E be a real Banach space and let T > 0 be fixed
We show that a well-known result on existence of mild solutions in Hilbert spaces can be extended to non-autonomous Hamilton-Jacobi equations in Banach spaces
The object of this paper is to study the existence of a mild solution V: [0, T ] × E → R to the following final-value problem for the non-autonomous semi-linear HamiltonJacobi partial differential equation (HJ-PDE) on [0, T ] × E
Summary
The object of this paper is to study the existence of a mild solution V: [0, T ] × E → R to the following final-value problem for the non-autonomous semi-linear HamiltonJacobi partial differential equation (HJ-PDE) on [0, T ] × E,. For the case in which E is a Hilbert space and Equation (1.1) is autonomous with respect to time variable (i.e. A(t) and G(t) do not depend on t), this regularizing property has been used in conjunction with a fixed point argument to prove existence of a unique solution to the integral Equation (1.2) in a certain space of functions, see e.g. Theorem 9.3 in Zabczyk (1999, Sec. 9), Da Prato and Zabczyk (2002, Part III) and Masiero (2005). We prove the transition operators of the (mild) solution verify the assumptions of the main result This leads to our final Example 6.7
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