We study the asymptotic behavior of a semidiscrete numerical approximation for a pair of heat equations ut = ∆u, vt = ∆v in Ω × (0, T ); fully coupled by the boundary conditions ∂u ∂η = up11vp12 , ∂v ∂η = up21vp22 on ∂Ω× (0, T ), where Ω is a bounded smooth domain in Rd. We focus in the existence or not of non-simultaneous blow-up for a semidiscrete approximation (U, V ). We prove that if U blo...

In this paper, we study the solution of an initial boundary value problem for a quasilinear parabolic equation with a nonlinear boundary condition. We first show that any positive solution blows up in finite time. For a monotone solution, we have either the single blow-up point on the boundary, or blow-up on the whole domain, depending on the parameter range. Then, in the single blow-up point c...

There are many nonlinear parabolic equations whose solutions develop singularity in finite time, say T. In many cases, a certain norm of the solution tends to infinity as time t approaches T. Such a phenomenon is called blow-up, and T is called the blow-up time. This paper is concerned with approximation of blow-up phenomena in nonlinear parabolic equations. For numerical computations or for ot...

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies, then the resulting graph is called a balanced blow-up. We show that any graph which contains the maximum number of induced copies of a sufficiently large balan...

In this paper, we establish the local existence and uniqueness of the solution for the degenerate parabolic equation with a nonlocal source and homogeneous Dirichlet boundary condition. Moreover, we prove that the solution blows up in finite time and obtain the blow-up set in some special case. Mathematics Subject Classification: 35K20, 35K30, 35K65

We study finite blow-up solutions of the heat equation with nonlinear boundary conditions. We provide a sufficient condition for the single point blow-up at the origin and a precise spacial singularity of the blow-up profile. Mathematics subject classification (2010): 35K20, 35B44.

and Applied Analysis 3 Theorem 1.2. Assume that u0 x ∈ C1 0, ∞ is a nonnegative, nonincreasing and compactly supported function, then all the nontrivial solutions u x, t of problem 1.5 occur blow-up; moreover, the blow-up set B u satisfies [ 0, p ( p − 1) p − 2 ) ⊂ B u ⊂ [ 0, p ( p − 1) p − 2 ] . 1.6 Remark 1.3. The nonincreasing assumption on u0 makes the proof much simpler see also 7 . Remark...

‎In this paper‎, ‎we consider the existence and uniqueness of the global solution for the sixth-order damped Boussinesq equation‎. ‎Moreover‎, ‎the finite-time blow-up of the solution for the equation is investigated by the concavity method‎.

The temperature of a combustible material will rise or even blow up when a heat source moves across it. In this paper, we study the blow-up phenomenon in this kind of moving heat source problems in two-dimensions. First, a two-dimensional heat equation with a nonlinear source term is introduced to model the problem. The nonlinear source is localized around a circle which is allowed to move. By ...