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...

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we d...

Let X be a variety over an algebraically closed field K . Its Nash blow-up is a variety over K with a projective morphism to X , which is an isomorphism over the smooth locus. Roughly speaking, it parametrizes all limits of tangent planes to X (a precise definition is given in §2 below). The Nash blow-up of a singular X is not always smooth but seems, in some sense, to be less singular than X ....

The aim of this paper is to refine some results concerning the blow-up of solutions of the exponential reaction-diffusion equation. We consider solutions that blow-up in finite time, but continue to exist as weak solutions beyond the blow-up time. The main result is that these solutions become regular immediately after the blow-up time. This result improves on that of Fila, Matano and Polácik, ...

We consider the blow-up of solutions for a semilinear reaction diffusion equation with exponential reaction term. It is know that certain solutions that can be continued beyond the blow-up time possess a nonconstant selfsimilar blow-up profile. Our aim is to find the final time blow-up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach w...

We obtain some sufficient conditions under which solutions to a nonlinear parabolic equation of second order with nonlinear boundary conditions tend to zero or blow up in a finite time. We also give the asymptotic behavior of solutions which tend to zero as t→ ∞. Finally, we obtain the asymptotic behavior near the blow-up time of certain blow-up solutions and describe their blow-up set.