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Stabilized quasi-reversibility method for a class of nonlinear ill-posed problems. (English) Zbl 1171.35485

Summary: We study a final value problem for the nonlinear parabolic equation \[ \displaylines{ u_t+Au =h(u(t),t),\quad 0<t<T\cr u(T)= \varphi , } \] where \(A\) is a non-negative, self-adjoint operator and \(h\) is a Lipschitz function. Using the stabilized quasi-reversibility method presented by Miller, we find optimal perturbations of the operator \(A\) depending on a small parameter \(\epsilon \) to setup an approximate nonlocal problem. We show that the approximate problems are well-posed under certain conditions and that their solutions converges if and only if the original problem has a classical solution. We also obtain estimates for the solutions of the approximate problems, and show a convergence result. This paper extends work by B. M. C. Hetrick and R. J. Hughes [J. Math. Anal. Appl. 331, No. 1, 342–357 (2007; Zbl 1127.34029)] to nonlinear ill-posed problems.

MSC:

35R25 Ill-posed problems for PDEs
35K05 Heat equation
47J06 Nonlinear ill-posed problems
47H10 Fixed-point theorems
35A35 Theoretical approximation in context of PDEs
35B45 A priori estimates in context of PDEs

Citations:

Zbl 1127.34029
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