Deduction of L. Hörmander's extension of Ásgeirsson's mean value theorem

L. Hörmander's extension of Ásgeirsson's mean value theorem states that if u is a solution of the inhomogeneous ultrahyperbolic equation (Δ_x−Δ_y)u=f, , , then

     

Weighted weak type inequalities for modified Hardy operators and geometric means operators in dimensions one and greater

We characterize the pairs of weights (u,v) such that the geometric mean operator G₁, defined for positive functions f on (0,∞) by

     

Regularity criterion of weak solutions to the 3D Navier-Stokes equations via two velocity components

This paper is concerned with the regularity criterion of Leray-Hopf weak solutions to the 3D Navier-Stokes equations with respect to Serrin type condition on two velocity filed components. It is shown that the weak solution u=(u₁,u₂,u₃) is regular on (0,T] if there exist two solution components, for example, u₂ and u₃, satisfying the condition

     

Relaxation and regularity in the calculus of variations

In this work we prove that, if L(t,u,ξ) is a continuous function in t and u, Borel measurable in ξ, with bounded non-convex pieces in ξ, then any absolutely continuous solution to the variational problem

     

Multiplicity of solutions for the plasma problem in two dimensions

Let Ω be a bounded domain in R², u₊=u if u?0, u₊=0 if u<0, u₋=u₊−u. In this paper we study the existence of solutions to the following problem arising in the study of a simple model of a confined plasma

     

On blow-up at space infinity for semilinear heat equations

A nonnegative blowing up solution of the semilinear heat equation ut=Δu+up with p>1 is considered when initial data u₀ satisfies

     

A complete classification of bifurcation diagrams of a p-Laplacian Dirichlet problem

We study the bifurcation diagrams of classical positive solutions u with ‖u_‖∞∈(0,∞) of the p-Laplacian Dirichlet problem

     

On positive solutions for some semilinear periodic parabolic eigenvalue problems

For a bounded domain Ω in , N?2, satisfying a weak regularity condition, we study existence of positive and T-periodic weak solutions for the periodic parabolic problem Lu_λ=λg(x,t,u_λ) in , u_λ=0 on . We characterize the set of positive eigenvalues with positive eigenfunctions associated, under the assumptions that g is a Caratheodory function such that ξ→g(x,t,ξ)/ξ is nonincreasing in (0,∞) a.e. satisfying some integrability conditions in (x,t) and

     

Localization for the evolution p-Laplacian equation with strongly nonlinear source term

In this paper we study the strict localization for the p-Laplacian equation with strongly nonlinear source term. Let u:=u(x,t) be a solution of the Cauchy problem

     

The Toda system and multiple-end solutions of autonomous planar elliptic problems

We prove the existence of a new class of entire, positive solutions for the classical elliptic problem Δu−u+up=0 in R², when p>2. The solutions we construct are obtained by perturbing the function

     

Initial blow-up of solutions of semilinear parabolic inequalities

We study classical nonnegative solutions u(x,t) of the semilinear parabolic inequalities

     

Global Hölder continuity of weak solutions to quasilinear divergence form elliptic equations

We derive global Hölder regularity for the -weak solutions to the quasilinear, uniformly elliptic equation

div(aij(x,u)Dju+ai(x,u))+a(x,u,Du)=0     

Existence, uniqueness, and quenching properties of solutions for degenerate semilinear parabolic problems with second boundary conditions

Let q?0, p?0, T?∞, D=(0,a), , Ω=D×(0,T), and Lu=xqut−uxx. This article considers the following degenerate semilinear parabolic initial-boundary value problem,

     

Generation, propagation, and annihilation of metastable patterns

We study the full-time dynamics of the initial value problem, for u^ε=u^ε(x,t),

     

Classification of Razor Blades to the filtration equation—the sublinear case

We study nonnegative solutions of the filtration equation u_t=Δ?(u) in , where ? is continuous, increasing and sublinear. More precisely, we study the Razor Blades, i.e., solutions which may be singular at |x|=0 for t>0, and start with zero initial data. We first prove a nonexistence result when ? is too sublinear and we show an axial trace result in the other case: there exist a time τ≡τ(u) and a Radon measure λ on [0,τ) such that

     

A generalized eigenproblem for the Laplacian which arises in lightning

The following generalized eigenproblem is analyzed: Find , u≠0, and λ∈R such that

D〈∇u,∇v〉=λΩ〈∇u,∇v〉     

Optimal bounds on the Kuramoto-Sivashinsky equation

In this paper, we consider solutions u(t,x) of the one-dimensional Kuramoto-Sivashinsky equation, i.e.

     

Global classical solution of Muskat free boundary problem

In this paper the Muskat problem which describes a two-phase flow of two fluids, for example, oil and water, in porous media is discussed. The problem involves in seeking two time-dependent harmonic functions u₁(x,y,t) and u₂(x,y,t) in oil and water regions, respectively, and the interface between oil and water, i.e., the free boundary Γ:y=ρ(x,t), such that on the free boundary