in this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,t)$, subject to nulldirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. the optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...

‎This note deals with the systems of parabolic equations with local and localized sources involving $n$ components‎. ‎We obtained the exponent regions‎, ‎where $kin {1,2,cdots,n}$ components may blow up simultaneously while the other $(n-k)$ ones still remain bounded under suitable initial data‎. ‎It is proved that different initial data can lead to different blow-up phenomena even in the same ...

In this paper, we consider singular and degenerate parabolic equations$$u_t =(x^alpha u_x)_x +u^m (x_0,t)v^{n} (x_0,t),quadv_t =(x^beta v_x)_x +u^q (x_0,t)v^{p} (x_0,t),$$ in $(0,a)times (0,T)$, subject to nullDirichlet boundary conditions, where $m,n, p,qge 0$, $alpha, betain [0,2)$ and $x_0in (0,a)$. The optimal classification of non-simultaneous and simultaneous blow-up solutions is determin...

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...

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution...

In this paper, we study positive blow-up solutions of the semilinear parabolic system with localized reactions ut = Δu+ vr + up(0,t), vt = Δv + us + vq(0,t) in the ball B = {x ∈ R N : |x| < R} , under the homogeneous Dirichlet boundary condition. It is shown that nonsimultaneous blow-up may occur according to the value of p , q , r , and s ( p,q,r,s > 1). We also investigate blow-up rates of al...