This paper deals with blow-up properties of solutions to a semilinear parabolic system with nonlinear localized source involved a product with local terms ut = Δu+ exp{mu(x,t)+nv(x0 ,t)}, vt = Δv+ exp{pu(x0,t)+qv(x,t)} with homogeneous Dirichlet boundary conditions. We investigate the influence of localized sources and local terms on blow-up properties for this system, and prove that: (i) when ...

This thesis is concerned with the study of the Blow-up phenomena for parabolic problems, which can be defined in a basic way as the inability to continue the solutions up to or after a finite time, the so called blow-up time. Namely, we consider the blow-up location in space and its rate estimates, for special cases of the following types of problems: (i) Dirichlet problems for semilinear equat...

We study the asymptotic behaviour of nonnegative solutions of the nonlinear diffusion equation in the half-line with a nonlinear boundary condition,    ut = uxx − λ(u + 1) log(u + 1) (x, t) ∈ R+ × (0, T ), −ux(0, t) = (u + 1) log(u + 1)(0, t) t ∈ (0, T ), u(x, 0) = u0(x) x ∈ R+, with p, q, λ > 0. We describe in terms of p, q and λ when the solution is global in time and when it blows up in f...

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...

lim t→T ‖u(t)‖H 1 0 ( ) =+∞. A point a ∈ is called a blow-up point of u if there exists (an, tn) → (a,T ) such that |u(an, tn)| → +∞. The set of all blow-up points of u(t) is called the blow-up set and denoted by S. From Giga and Kohn [8, Theorem 5.3], there are no blow-up points in ∂ . Therefore, we see from (3) and the boundedness of that S is not empty. Many papers are concerned with the Cau...

We analyze the blow-up behavior of one-parameter collocation solutions for Hammerstein-type Volterra integral equations (VIEs) whose solutions may blow up in finite time. To approximate such solutions (and the corresponding blow-up time), we will introduce an adaptive stepsize strategy that guarantees the existence of collocation solutions whose blow-up behavior is the same as the one for the e...

Formation of blow-up singularities for the Navier–Stokes equations (NSEs) ut + (u · ∇)u = −∇p+∆u, divu = 0 in R × R+, with bounded data u0 is discussed. Using natural links with blow-up theory for nonlinear reaction-diffusion PDEs, some possibilities to construct special self-similar and other related solutions that are characterized by blow-up swirl with the angular speed near the blow-up time...

Blow-up phenomena for solutions of some nonlinear parabolic systems with time dependent coefficients are investigated. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.