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.

In this paper we investigate a nonlocal reaction diffusion equation with potential, under Neumann boundary. We obtain the complete classification of the parameters for which the solution blows up in finite time or exists globally. Moreover, we study the blowup rate and the blowup set for the blowup solution. Key–Words: Nonlocal diffusion, Blow up, Blowup rate, Blowup set