Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source

Abstract

Let H be a Hilbert space with the inner product •, • and the norm • , A(t) a positive self-adjoint unbounded time-dependent operator on H and ε > 0. We establish stability estimates of Hölder type and propose a regularization method with error estimates of Hölder type for the ill-posed backward semilinear parabolic equationwith the source function f satisfying a local Lipschitz condition.

Convergent numerical methods for parabolic equations with reversed time via a new Carleman estimate

“…We now briefly compare results of this paper with the ones of the paper of Hào et al [9]. In [9] the problem with the time reversed data is considered for the quasilinear equation u t (t) + A (t) u (t) = f (t, u (t)) , where A (t) is a positive self-adjoint unbounded operator in a Hilbert space H. This is close to the problem we consider here, although we do not assume that our operator L in (1.5) is self-adjoint. The function f (t, u (t)) in [9] satisfies the local Lipschitz condition with respect to u, which is more general than our global Lipschitz condition (1.6).…”

“…In [9] the problem with the time reversed data is considered for the quasilinear equation u t (t) + A (t) u (t) = f (t, u (t)) , where A (t) is a positive self-adjoint unbounded operator in a Hilbert space H. This is close to the problem we consider here, although we do not assume that our operator L in (1.5) is self-adjoint. The function f (t, u (t)) in [9] satisfies the local Lipschitz condition with respect to u, which is more general than our global Lipschitz condition (1.6). The applied importance of the local Lipschitz condition is demonstrated in [9] using some specific examples of parabolic PDEs arising in Physics.…”

Landweber–Kaczmarz for parameter identification in time-dependent inverse problems: all-at-once versus reduced version

“…• The local Lipschitz condition (A2) is weaker than the one used in [10], as we only have the V * -norm on the left hand side of the Lipschitz condition. • Note that differentiablity of F can be interpreted on the stronger image space C(0, T; V * ) since r(.)…”

“…The main assumption we rely on is basically the local Lipschitz continuity condition. Recently, some authors [10,24] also employ local Lipschitz continuity as the key ingredient for the research on backward parabolic problems, but rather in a semigroup framework than in the Sobolev space framework, e.g. of [22], which we rely on to get even somewhat weaker conditions.…”

On existence and regularity of a terminal value problem for the time fractional diffusion equation

“…Wei and her group studied some regularization methods for homogeneous backward problem and Yamamoto et al [53] considered a backward problem in time for a time-fractional partial differential equation in the one-dimensional case. When α = 1, systems (1.1)-(1.3) are reduced to the backward problem for classical reaction diffusion equations, and were studied in [54][55][56].…”