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.