In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on (0, T ) × Ω :where Ω is a bounded domain in R n and λ, γ > 0. In this work, we have succeeded to construct a solution which quenches in finite time T only at one interior point a ∈ Ω. In particular, we give a description of the quenching behavior according to the following final profile1 3 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.
We consider the semilinear heat equationin the whole space ޒ n , where p > 1 and α ∈ .ޒ Unlike the standard case α = 0, this equation is not scaling invariant. We construct for this equation a solution which blows up in finite time T only at one blowup point a, according to the asymptotic dynamicwhere ψ(t) is the unique positive solution of the ODEThe construction relies on the reduction of the problem to a finite-dimensional one and a topological argument based on the index theory to get the conclusion. By the interpretation of the parameters of the finite-dimensional problem in terms of the blowup time and the blowup point, we show the stability of the constructed solution with respect to perturbations in initial data. To our knowledge, this is the first successful construction for a genuinely non-scale-invariant PDE of a stable blowup solution with the derivation of the blowup profile. From this point of view, we consider our result as a breakthrough.
In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem: [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary [Formula: see text] such problem is derived as the shadow limit of a singular Gierer–Meinhardt system, Kavallaris and Suzuki [On the dynamics of a non-local parabolic equation arising from the Gierer–Meinhardt system, Nonlinearity (2017) 1734–1761; Non-Local Partial Differential Equations for Engineering and Biology: Mathematical Modeling and Analysis, Mathematics for Industry, Vol. 31 (Springer, 2018)]. Under the Turing type condition [Formula: see text] we construct a solution which blows up in finite time and only at an interior point [Formula: see text] of [Formula: see text] i.e. [Formula: see text] where [Formula: see text] More precisely, we also give a description on the final asymptotic profile at the blowup point [Formula: see text] and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity 10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci. 29 (2019) 1279–1348].
In this paper, we consider the following semi-linear complex heat equation ∂tu = ∆u + u p , u ∈ C in R n , with an arbitrary power p, p > 1. In particular, p can be non integer and even irrational, unlike our previous work [5], dedicated to the integer case. We construct for this equation a complex solution u = u 1 + iu 2 , which blows up in finite time T and only at one blowup point a. Moreover, we also describe the asymptotics of the solution by the following final profiles:2) This model is connected to the viscous Constantin-Lax-Majda equation with a viscosity term, which is a one dimensional model for the vorticity equation in fluids. For more details, the readers are addressed to the following works: Constantin, Lax, Majda [2], Guo, Ninomiya and Yanagida in [8], Okamoto, Sakajo and Wunsch [25], Sakajo in [26] and [27], Schochet [28]. In [5], we treated the case p ∈ N. Indeed, handling the 2010 Mathematics Subject Classification. Primary: 35K50, 35B40; Secondary: 35K55, 35K57.
In this paper, we aim to refine the blow-up behavior for the complex Ginzburg-Landau (CGL) equation in some subcritical case. More precisely, we construct blow-up solutions and refine their blow-up profile to the next order.
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