We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains O ⊆ Rd in the scale Bα τ,τ (O), 1/τ = α/d + 1/p, p ≥ 2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing...

In this note, we study the semilinear wave equation with power nonlinearity $|u|^p$ on compact Lie groups. First, prove a local in time existence result energy space via Fourier analysis Then, blow-up for Cauchy problem any $p>1$, under suitable sign assumptions initial data. Furthermore, sharp lifespan estimates (in time) solutions are derived.

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 boundary blow-up solutions of semilinear elliptic equations Lu = up + with p > 1, or Lu = e with a > 0, where L is a second order elliptic operator with measurable coefficients. Several uniqueness theorems and an existence theorem are obtained.

We consider the following eigenvalue problem: −Δu f u λu, u u x , x ∈ B {x ∈ R3 : |x| < 1}, u 0 p > 0, u||x| 1 0, where p is an arbitrary fixed parameter and f is an odd smooth function. First, we prove that for each integer n ≥ 0 there exists a radially symmetric eigenfunction un which possesses precisely n zeros being regarded as a function of r |x| ∈ 0, 1 . For p > 0 sufficiently small, such...

and Applied Analysis 3 However we have, for the complex solution,