Corrigendum to “Lower bounds for the blow-up time in the higher-dimensional nonlinear divergence form parabolic equations” [C. R. Acad. Sci. Paris, Ser. I 351 (19–20) (2013) 731–735] (original) (raw)
Upper bounds for the blow up time for the Kirchhoff- type equation
Communications Faculty of Sciences University of Ankara. Series A1: mathematics and statistics, 2023
In this research, we take into account the Kirchhoff type equation with variable exponent. The Kirchhoff type equation is known as a kind of evolution equations,namely, PDEs, where t is an independent variable. This type problem can be extensively used in many mathematical models of various applied sciences such as flows of electrorheological fluids, thin liquid films, and so on. This research, we investigate the upper bound for blow up time under suitable conditions.
Blow-up Rate Estimates for Parabolic Equations
We consider the blow-up sets and the upper blow-up rate estimates for two parabolic problems defined in a ball B R in R n ; firstly, the semilinear heat equation u t = ∆u + e u p subject to the zero Dirichlet boundary conditions, secondly, the problem of the heat equation u t = ∆u with the Neumann boundary condition ∂u ∂η = e u p on ∂B R × (0, T), where p > 1, η is the outward normal.
Bounds for blow-up time in a semilinear parabolic problem with viscoelastic term
Computers & Mathematics with Applications, 2017
We consider a semilinear parabolic equation with viscoelastic term u t − △u + ∫ t 0 g(t − s)△u(x, s)ds = |u| p−2 u. By the means of differential inequality technique, we obtain a lower bound for blow-up time of the solution if blow-up occurs. At the same time, we establish a new blow-up criterion and give an upper bound for blow-up time of the solution under some conditions on p, g and u 0 .
Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities
Colloquium Mathematicum, 2001
Consider the nonlinear heat equation (E): u t − ∆u = |u| p−1 u + b|∇u| q. We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C 1 (T − t) −1/(p−1) ≤ u(t) ∞ ≤ C 2 (T − t) −1/(p−1). Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality u t − u xx ≥ u p. More general inequalities of the form u t − u xx ≥ f (u) with, for instance, f (u) = (1 + u) log p (1 + u) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary differential inequalityv ≥ f (v).
An introduction to the problem of blow-up for semilinear and quasilinear parabolic equations
MAT Serie A
In this article we present an overview of the mathematical problem of blow-up for semilinear parabolic equations. We describe in a simple model the main questions posed in the theory: which solutions blow up, where and how they do. We also give some introduction to the methods and tools used in the proofs. Related problems, like quasilinear equations or blow-up produced by a boundary flux, are also treated. Resumen. En este artículo se presenta una panorámica del problema matemático de explosión en ecuaciones parabólicas semilineales, describiendo las principales técnicas usadas generalmente en este tipo de problemas. Con la ayuda del ejemplo clásico de la propagación del calor con reacción se estudian las principales cuestiones de la teoría, incluyendo qué soluciones explotan, dónde y cómo lo hacen. Finalmente se incluyen algunos problemas relacionados, como el caso de ecuaciones quasilineales, sistemas o el de explosión producida por reacciones frontera.
Blowup rate estimates for the heat equation with a nonlinear gradient source term
Discrete and Continuous Dynamical Systems, 2008
The gradient blowup rate of the equation u t = ∆u + |∇u| p , where p > 2, is studied. It is shown that the blowup rate will never match that of the self-similar variables. In the one space dimensional case when assumptions are made on the initial data so that the solution is monotonically increasing in time, the exact blowup rate is found.
Numerical Blow-Up Time with Respect to Parameters for a Reaction Diffusion Equation
Journal of Mathematical Sciences: Advances and Applications, 2019
some conditions under which the solution of semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also prove the convergence of the semidiscrete form blow-up time to the real one when the mesh size tends to zero. Finally, we give some numerical results to illustrate our analysis.
arXiv (Cornell University), 2023
In this paper, we obtain upper bounds for the critical time T * of the blowup for the parabolic-elliptic Patlak-Keller-Segel system on the 2D-Euclidean space. No moment condition or/and entropy condition are required on the initial data; only the usual assumptions of non-negativity and finiteness of the mass are assumed. The result is expressed not only in terms of the supercritical mass M > 8π, but also in terms of the shape of the initial data. 1 Contents 1. General upper bound on T * for the (PKS) system 1 2. Estimates on the critical time bound T * c (n 0) 11 3. Lower bounds on T * c (n 0) 25 4. Examples of initial data 29 4.1. Gaussian initial data n 0 29 4.2. Characteristic function of a disk 34 4.3. More examples of initial data 38 References 39
Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations
Journal of Dynamics and Differential Equations, 2007
We consider reaction diffusion equations of the prototype form u t = u xx + λu + |u| p−1 u on the interval 0 < x < π, with p > 1 and λ > m 2 . We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included.
The blow-up problem for a semilinear parabolic equation with a potential
Journal of Mathematical Analysis and Applications, 2007
Let Ω be a bounded smooth domain in R N . We consider the problem u t = ∆u + V (x)u p in Ω × [0, T ), with Dirichlet boundary conditions u = 0 on ∂Ω × [0, T ) and initial datum u(x, 0) = M u 0 (x) where M ≥ 0, u 0 is positive and compatible with the boundary condition. We give estimates for the blow up time of solutions for large values of M . As a consequence of these estimates we find that, for M large, the blow up set concentrates near the points where u p−1 0 V attains its maximum.