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...

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.

The purpose of this work is to deal with the blow-up behavior of the nonnegative solution to a degenerate and singular parabolic equation with nonlocal boundary condition. The conditions on the existence and non-existence of the global solution are given. Further, under some suitable hypotheses, we discuss the blow-up set and the uniform blow-up profile of the blow-up solution. c ©2016 All righ...

We study solutions of some supercritical parabolic equations which blow up in finite time but continue to exist globally in the weak sense. We show that the minimal continuation becomes regular immediately after the blow-up time and if it blows up again, it can only do so finitely many times.

We consider operations that change the size of images, either shrinks or blow-ups. Image processing offers numerous possibilities, put at everyone's disposal with such computer programs as Adobe Photoshop. We consider a different class of operations, aimed at immediate visual awareness, rather than pixel arrays. We demonstrate cases of blow-ups that do not sacrifice apparent resolution. This ap...

The blow-up of solutions for a class of quasilinear reaction-diffusion equations with a gradient term ut = div(a(u)b(x)∇u)+ f (x,u, |∇u|2, t) under nonlinear boundary condition ∂u/∂n + g(u) = 0 are studied. By constructing a new auxiliary function and using Hopf’s maximum principles, we obtain the existence theorems of blow-up solutions, upper bound of blow-up time, and upper estimates of blow-...

The paper studies the finite-time blow-up theory for a class of nonlinear Volterra integro-differential equations. The conditions for the occurrence of finite-time blow-up for nonlinear Volterra integro-differential equations are provided. Moreover, the finite-time blow-up theory for nonlinear partial Volterra integro-differential equations with general kernels is also established using the blo...