Finite-time blow-up of a non-local stochastic parabolic problem

Abstract

In the current work we investigate the behaviour of a non-local stochastic parabolic problem. We first prove the local-in-time existence and uniqueness of a weak solution. Then we check the extendability in time of the weak solution. In particular, the rest of paper is devoted to the investigation of the conditions under which finitetime blow-up occurs. We first prove that noise term induced finite-time blow up takes place when the stochastic term is rather big independently of the size of the non-local term. Afterwards, non-local-term induced finite-time blow occurs when the nonlinearity satisfies some specific growth conditions. In this case some fundamental results like maximum principle and Hopf's type lemma in the context of SPDEs are first provided and later Kaplan's eigenfunction method adjusted to the our non-local SPDE is employed.

Impacts of noise on quenching of some models arising in MEMS technology

“…Thus, under some imposed uncertainty model (2.1) can be transformed to (1.1a). Notably, the choice θ = 3 is made since it leads to a linear type diffusion term for which case the local existence theory is well established for a general Lipschitz nonlinearity, cf [8,33], and [27] for the non-Lipschitz nonlinearity (1 − u) −2 . Also, for the case of a model with a general diffusion term σ (u) the interested reader can check [27].…”

“…Next, we consider the case of nonhomogeneous boundary conditions of the form (1.1b) or equivalently (4.2b) with β = β c , since such a case is of particular interest in the light of the quenching results of section 5. It is sufficient for Robin-type boundary conditions to be satisfied in the weak sense, although they could even hold in the classical sense too, see [33,Theorem 4.1]. A simulation implementing the previously described numerical algorithm for this O. Drosinou et al Additionally, in the following Table (T2) we present the results of such a numerical experiment.…”

Impacts of noise on quenching of some models arising in MEMS technology

“…and thus the above analysis reveals that under some imposed uncertainty model (2.1) can be transformed to (1.1a). Furthermore, it should be noted that the choice θ = 3 leads to a linear type diffusion term for which case the local existence theory is well established, cf [8,25,31]. For the case of a model with a general diffusion term σ(u) the interested reader can check [25].…”