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. … Show more
“…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].…”
Section: The Mathematical Modelmentioning
confidence: 99%
“…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.…”
In the current work, we study a stochastic parabolic problem. The presented problem is motivated by the study of an idealised electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference, a parameter that actually controls the operation of MEMS device. We first present the construction of the mathematical model, and then, we deduce some local existence results. Next for some particular versions of the model, relevant to various boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, which also allows the further investigation the problem with respect to its quenching behaviour.
“…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].…”
Section: The Mathematical Modelmentioning
confidence: 99%
“…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.…”
In the current work, we study a stochastic parabolic problem. The presented problem is motivated by the study of an idealised electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference, a parameter that actually controls the operation of MEMS device. We first present the construction of the mathematical model, and then, we deduce some local existence results. Next for some particular versions of the model, relevant to various boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, which also allows the further investigation the problem with respect to its quenching behaviour.
“…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].…”
In the current work we study a stochastic parabolic problem. The underlying problem is actually motivated by the study of an idealized electrically actuated MEMS (Micro-Electro-Mechanical System) device in the case of random fluctuations of the potential difference controlling the device. We first present the mathematical model and then we deduce some local existence results. Next for some particular versions of the model, regarding its boundary conditions, we derive quenching results as well as estimations of the probability for such singularity to occur. Additional numerical study of the problem in one dimension follows, investigating the problem further with respect to its quenching behaviour.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.