Five types of blow-up patterns that can occur for the 4th-order semilinear parabolic equation of reaction-diffusion type ut = −∆2u+ |u|p−1u in R × (0, T ), p > 1, limt→T− supx∈RN |u(x, t)| = +∞, are discussed. For the semilinear heat equation ut = ∆u+ u , various blow-up patterns were under scrutiny since 1980s, while the case of higher-order diffusion was studied much less, regardless a wide r...

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 the present note, we address the question about behavior of L3-norm of the velocity field as time t approaches blow-up time T . It is known that the upper limit of the above norm must be equal to infinity. We show that, for blow-ups of type I, the lower limit of L3-norm equals to infinity as well. 1991 Mathematical subject classification (Amer. Math. Soc.): 35K, 76D.

Thermal blow-up in a subdiffusive medium with a localized energy source is examined for a spatial domain of infinite extent in one, two, and three dimensions. An analysis of a nonlinear model of this problem reveals that a blow-up always occurs, independent of the spatial dimension and the thermal properties of the material. This behavior is in contrast with both classical diffusion and superdi...

We consider a special type of a hyperbolic system and show that classical solutions blow up in finite time even for small initial data. 2000 Mathematics Subject Classification. 35L45.

In this paper we investigate results on finite-time blow-up of solutions to a nonlocal quasi-linear parabolic problem. We extend some known results to higher dimensions. AMS subject classification: 35K55, 35K65, 35B35.

Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrödinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. It is white in time and colored in space. The path space considered allows blow-up and is endowed with a topology analogue to a projective limit topology. Thus a large variety of la...

In this paper we consider a kind of higher-order evolution equation as^{kt^{k} + ^{k&minus1}u/t^{k&minus1} +• • •+ut &minus{delta}u= f (u, {delta}u,x). For this equation, we investigate nonglobal solution, blow-up in finite time and instantaneous blow-up under some assumption on k, f and initial data. In this paper we employ the Test function method, the eneralized convexity method an...