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Keyword: Semilinear Heat Equation (5)	28 Mar 2025	Run
[in Journal: Advances in Synchrotron Radiation] AND [Keyword: Thin Films] (1)	28 Mar 2025	Run
Keyword: Functional Data Analysis (11)	28 Mar 2025	Run
[in Journal: Biophysical Reviews and Letters] AND [Keyword: Time Delay] (1)	28 Mar 2025	Run
Keyword: Δ-unknotting Number (4)	28 Mar 2025	Run

articleNo Access
Profile of a touch-down solution to a nonlocal MEMS model
- Giao Ky Duong and
- Hatem Zaag
Mathematical Models and Methods in Applied Sciences24 Jun 2019
Preview Abstract
In this paper, we are interested in the mathematical model of MEMS devices which is presented by the following equation on $(0, T) \times Ω$ $(0, T) \times Ω$ :
$\partial_{t} u = Δ u + \frac{λ}{{(1 - u)}^{2} {(1 + γ \int_{Ω} \frac{1}{1 - u} d x)}_{Ω}^{2}}, 0 \leq u < 1,$ $\partial_{t} u = Δ u + \frac{λ}{{(1 - u)}^{2} {(1 + γ \int_{Ω} \frac{1}{1 - u} d x)}^{2}}, 0 \leq u < 1,$
where $Ω$ $Ω$ 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 $a \in Ω$ . In particular, we give a description of the quenching behavior according to the following final profile
$1 - u (x, T) \sim 𝜃^{*} {[\frac{| x - a |^{2}}{| ln | x - a | |}]}^{\frac{1}{3}} as x \to a, 𝜃^{*} > 0 .$
The construction relies on some connections between the quenching phenomenon and the blowup phenomenon. More precisely, we change our problem to the construction of a blowup solution for a related PDE and describe its behavior. The method is inspired by the work of Merle and Zaag [Reconnection of vortex with the boundary and finite time quenching, Nonlinearity10 (1997) 1497–1550] with a suitable modification. In addition to that, the proof relies on two main steps: A reduction to a finite-dimensional problem and a topological argument based on index theory. The main difficulty and novelty of this work is that we handle the nonlocal integral term in the above equation. The interpretation of the finite-dimensional parameters in terms of the blowup point and the blowup time allows to derive the stability of the constructed solution with respect to initial data.
articleNo Access
ON EXACT DEAD-CORE RATES FOR A SEMILINEAR HEAT EQUATION WITH STRONG ABSORPTION
- YUKIHIRO SEKI
Communications in Contemporary Mathematics01 Feb 2011
Preview Abstract
We study a semilinear heat equation with strong absorption u_t = Δu - u^p with 0 < p < 1 in R^N. A solution is known to develop dead-core in finite time for a wide class of initial data. We construct specific solutions with exact dead-core rates faster than the one given by the corresponding ODE. They are constructed formally by means of matched asymptotic expansion technique and rigorously by means of topological fixed-point arguments based on a priori estimates. To obtain the a priori estimates, we analyze a certain linearized problem in a new function space .
articleNo Access
SINGULAR SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR ABSORPTION AND HARDY POTENTIALS
- VITALI LISKEVICH,
- ANDREY SHISHKOV, and
- ZEEV SOBOL
Communications in Contemporary Mathematics01 Apr 2012
Preview Abstract
We study the existence and nonexistence of singular solutions to the equation , p > 1, in ℝ^N × [0, ∞), N ≥ 3, with a singularity at the point (0, 0), that is, nonnegative solutions satisfying u(x, 0) = 0 for x ≠ 0, assuming that α > -2 and . The problem is transferred to the one for a weighted Laplace–Beltrami operator with a nonlinear absorption, absorbing the Hardy potential in the weight. A classification of a singular solution to the weighted problem either as a source solution with a multiple of the Dirac mass as initial datum, or as a unique very singular solution, leads to a complete classification of singular solutions to the original problem, which exist if and only if .
articleNo Access
Blow up of solutions of semilinear heat equations in general domains
Communications in Contemporary Mathematics17 Feb 2015
Preview Abstract
Consider the nonlinear heat equation v_t - Δv = |v|^p-1v in a bounded smooth domain Ω ⊂ ℝⁿ with n > 2 and Dirichlet boundary condition. Given u_p a sign-changing stationary classical solution fulfilling suitable assumptions, we prove that the solution with initial value ϑu_p blows up in finite time if |ϑ - 1| > 0 is sufficiently small and if p is sufficiently close to the critical exponent . Since for ϑ = 1 the solution is global, this shows that, in general, the set of the initial data for which the solution is global is not star-shaped with respect to the origin. This phenomenon had been previously observed in the case when the domain is a ball and the stationary solution is radially symmetric.
chapterNo Access
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF SOME SEMILINEAR HEAT EQUATIONS IN R^N
- KAZUHIRO ISHIGE and
- TATSUKI KAWAKAMI
Recent Advances in Nonlinear Analysis01 Feb 2008
Preview Abstract
We consider the Cauchy problem of the semilinear heat equation,
where N ≥ 1, f ∈ C¹([0,∞)), and ϕ ∈ L¹(R^N) ∩ L^∞(R^N) satisfying some suitable conditions. We study the asymptotic behavior of the solutions in the L^q space for q ∈ [1,∞] by using the relative entropy methods.