Blow-up with logarithmic nonlinearities

Blow-up with logarithmic nonlinearities

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...

The Blow-up Problem for Exponential Nonlinearities

We give a solution of the blow-up problem for equation u = e, with data close to constants, in any number of space dimensions: there exists a blow-up surface, near which the solution has logarithmic behavior; its smoothness is estimated in terms of the smoothness of the data. More precisely, we prove that for any solution of u = e with Cauchy data on t = 1 close to (ln 2; 2) in H(R) H (R), s is...

From blow-up boundary solutions to equations with singular nonlinearities

In this survey we report on some recent results related to various singular phenomena arising in the study of some classes of nonlinear elliptic equations. We establish qualitative results on the existence, nonexistence or the uniqueness of solutions and we focus on the following types of problems: (i) blow-up boundary solutions of logistic equations; (ii) Lane-Emden-Fowler equations with singu...

Boundary Blow-Up Solutions to p x -Laplacian Equations with Exponential Nonlinearities

Blow-up for a non-Newtonian Polytropic Filtration Equation with Multiple Nonlinearities

This paper deals with the global existence and blow-up of a non-Newtonian polytropic filtration equation with multiple nonlinearities. We obtain necessary and sufficient conditions on the global existence of all positive solutions by constructing super solutions and sub solutions. AMS Subject Classifications: 35K50, 35K55, 35K65.