We study the numerical solution of semilinear parabolic PDEs on unbounded spatial domains in R2 whose solutions blow up in finite time. Of particular interest are the cases where = R2 or is a sectorial domain in R2. We derive the nonlinear absorbing boundary conditions for corresponding, suitably chosen computational domains and then employ a simple adaptive time-stepping scheme to compute the ...

Some oscillation criteria for the second order semilinear elliptic differential equation

We obtain sharp pointwise estimates for positive solutions to the equation −Lu+V u = f , where L is an elliptic operator in divergence form, q ∈ R\{0}, f ≥ 0 and V is a function that may change sign, in a domain Ω in R, or in a weighted Riemannian manifold.

We show that large positive solutions exist for the equation (P±) :∆u±|∇u|q = p(x)uγ in Ω ⊆ RN(N ≥ 3) for appropriate choices of γ > 1,q > 0 in which the domain Ω is either bounded or equal to RN . The nonnegative function p is continuous and may vanish on large parts of Ω. If Ω = RN , then p must satisfy a decay condition as |x| →∞. For (P+), the decay condition is simply ∫∞ 0 tφ(t)dt <∞, wher...

We study the semilinear wave equation in Schwarzschild metric (3 + 1 dimensional space time). First, we establish that the problem is locally well posed in H for any σ > 1; then we prove the blow up of the solution for every p > 1 and non negative initial data. The work is dedicated to prof. Yvonne Choquet Bruhat in occasion of her 80th year.

We study the decay of I (t) = R u(:; t) where u is a nonnegative solution to u t ? u + jruj q = 0 in R n R + with n 1. If 1 q n+2 n+1 then I (t) ! 0 if I (0) < 1 and inf t>0 I (t) > 0 if q > n+2 n+1 .

We study the nonexistence result of radial solutions to −∆u + c u |x|2 + |x| σ |u|u ≤ 0 posed in B or in B \ {0} where B is the unit ball centered at the origin in RN , N ≥ 3. Moreover, we give a complete classification of radial solutions to the problem −∆u + c u |x|2 + |x| σ |u|u = 0. In particular we prove that the latter has exactly one family of radial solutions.

Suucient conditions for controllability of semilinear integrodif-ferential systems in a Banach space are established. The results are obtained using the asymptotic xed-point theorem for k-set contractions .