Uniqueness and nondegeneracy for some nonlinear elliptic problems in a ball

In this paper we study the uniqueness and nondegeneracy of positive solutions of nonlinear problems of the type Δ_pu+f(r,u)=0 in the unit ball B, u=0 on ∂B. Here Δ_p denotes the p Laplace operator Δ_p=div(|∇u|^p−2∇u), p>1. The main ideas rely on the Maximum Principle and an implicit function theorem that we derive in a suitable weighted space. This space is essential to deal with the case p≠2.     

Study of weak solutions for parabolic equations with nonstandard growth conditions

The authors of this paper study the existence and uniqueness of weak solutions of the initial and boundary value problem for ut=div((uσ+d₀)|∇u|^p(x,t)−2∇u)+f(x,t). Localization property of weak solutions is also discussed.     

Global solutions of the fast diffusion equation with gradient absorption terms

We investigate the existence of nonnegative weak solutions to the problem ut=Δ(um)−p|∇u| in Rn×(0,∞) with ⁺(1−2/n)<m<1. It will be proved that: (i) When 1<p<2, if the initial datum u₀∈D(Rn) then there exists a solution; (ii) When 1<p<(2+mn)/(n+1), if the initial datum u₀(x) is a bounded and nonnegative measure then the solution exists; (iii) When (2+mn)/(n+1)?p<2, if the initial datum is a Dirac mass then the solution does not exist. We also study the large time behavior of the L¹-norm of solutions for 1<p?(2+mn)/(n+1), and the large time behavior of t1/β‖u(⋅,t)−Ec(⋅,t)L^∞_‖ for (2+mn)/(n+1)<p<2.     

Second critical exponent and life span for pseudo-parabolic equation

This paper studies the second critical exponent and life span of solutions for the pseudo-parabolic equation u_t−kΔu_t=Δu+u^p in Rⁿ×(0,T), with p>1, k>0. It is proved that the second critical exponent, i.e., the decay order of the initial data required by global solutions in the coexistence region of global and non-global solutions, is independent of the pseudo-parabolic parameter k. Nevertheless, it is revealed that the viscous term kΔu_t relaxes restrictions on the amplitude of the initial data required by the global solutions. Moreover, it is observed that the life span of the non-global solutions will be delayed by the third order viscous term. Finally, some numerical examples are given to illustrate all these results.     

Local regularity theorems for the stationary thermistor problem with oscillating degeneracy

In this paper we present some regularity results for solutions to the system −Δu=σ(u)²|∇φ|, div(σ(u)∇φ)=0 in the case where σ(u) is allowed to oscillate between 0 and a positive number as u→∞. In particular, we show that u is locally bounded if σ(u) is bounded below by a suitable exponential function.     

Positive solutions of Lane-Emden systems with negative exponents: Existence, boundary behavior and uniqueness

We study the existence, boundary behavior and uniqueness of solutions for the singular elliptic system −Δu=u^−pv^−q,−Δv=u^−rv^−s,u>0,v>0,x∈Ω,u|_∂Ω=v|_∂Ω=0, where Ω is a bounded domain with smooth boundary in R^N, p,s≥0 and q,r>0. Our results are obtained in a range of p,q,r,s different from those in [M. Ghergu, Lane-Emden systems with negative exponents, J. Funct. Anal. 258 (2010) 3295-3318].     

Formal solutions of semilinear heat equations

We study formal power series solutions to the initial value problem for semilinear heat equation t_∂u−Δu=f(u) with polynomial nonlinearity f and prove that they belong to the formal Gevrey class G². Next we give counterexamples showing that the solution, in general, is not analytic in time at t=0.     

Local existence of solutions to some degenerate parabolic equation associated with the p-Laplacian

The existence of local (in time) solutions of the initial-boundary value problem for the following degenerate parabolic equation: ut(x,t)−Δpu(x,t)−|u|^q−2u(x,t)=f(x,t), (x,t)∈Ω×(0,T), where 2?p<q<+∞, Ω is a bounded domain in RN, is given and Δp denotes the so-called p-Laplacian defined by Δpu:=∇⋅(|∇u|^p−2∇u), with initial data u₀∈Lr(Ω) is proved under r>N(q−p)/p without imposing any smallness on u₀ and f. To this end, the above problem is reduced into the Cauchy problem for an evolution equation governed by the difference of two subdifferential operators in a reflexive Banach space, and the theory of subdifferential operators and potential well method are employed to establish energy estimates. Particularly, Lr-estimates of solutions play a crucial role to construct a time-local solution and reveal the dependence of the time interval [0,T₀] in which the problem admits a solution. More precisely, T₀ depends only on Lr|u₀| and f.     

Global existence of solutions for semilinear damped wave equation in 2-D exterior domain

We consider a mixed problem of a damped wave equation u_tt−Δu+u_t=|u|^p in the two dimensional exterior domain case. Small global in time solutions can be constructed in the case when the power p on the nonlinear term |u|^p satisfies p∗=2<p<+∞. For this purpose we shall deal with a radially symmetric solution in the exterior domain. A new device developed in Ikehata-Matsuyama (Sci. Math. Japon. 55 (2002) 33) plays an effective role.     

Existence and multiplicity of solutions for quasilinear nonhomogeneous problems: An Orlicz-Sobolev space setting

We study the boundary value problem −div(log(1+q|∇u|)|∇u|^p−2∇u)=f(u) in Ω, u=0 on ∂Ω, where Ω is a bounded domain in RN with smooth boundary. We distinguish the cases where either f(u)=−λ|u|^p−2u+|u|^r−2u or f(u)=λ|u|^p−2u−|u|^r−2u, with p, q>1, p+q<min{N,r}, and r<(Np−N+p)/(N−p). In the first case we show the existence of infinitely many weak solutions for any λ>0. In the second case we prove the existence of a nontrivial weak solution if λ is sufficiently large. Our approach relies on adequate variational methods in Orlicz-Sobolev spaces.     

Isolated singularities of solutions to quasi-linear elliptic equations with absorption

We study the problem of removability of isolated singularities for a general second-order quasi-linear equation in divergence form −divA(x,u,∇u)+a₀(x,u)+g(x,u)=0 in a punctured domain Ω?{0}, where Ω is a domain in Rn, n?3. The model example is the equation −Δpu+gu|u|^p−2+u|u|^q−1=0, q>p−1>0, p<n. Assuming that the lower-order terms satisfy certain non-linear Kato-type conditions, we prove that for all point singularities of the above equation are removable, thus extending the seminal result of Brezis and Véron.     

Some degenerate and quasilinear parabolic systems not in divergence form

This paper deals with positive solutions of degenerate and quasilinear parabolic systems not in divergence form: u_t=u^p(Δu+av), v_t=v^q(Δv+bu), with null Dirichlet boundary conditions and positive initial conditions, where p, q, a and b are all positive constants. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that all solutions exist globally if and only if ab?λ₁², where λ₁ is the first eigenvalue of −Δ in Ω with homogeneous Dirichlet boundary condition.     

Blow-up problem for semilinear heat equation with absorption and a nonlocal boundary condition

In this paper we consider a semilinear parabolic equation u_t=Δu−c(x,t)u^p for (x,t)∈Ω×(0,∞) with nonlinear and nonlocal boundary condition u∣_{∂Ω×(0,∞)}=∫_Ωk(x,y,t)u^ldy and nonnegative initial data where p>0 and l>0. We prove some global existence results. Criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data are also given.     

Mathematical analysis to a nonlinear fourth-order partial differential equation

The paper first study the steady-state thin film type equation

∇⋅(uⁿ|∇Δu|^q−2∇Δu)−δu^mΔu=f(x,u)     

Existence and uniqueness of nonnegative solutions for a boundary blow-up problem

Assume that Ω is a bounded domain in RN (N?3) with smooth boundary ∂Ω. In this work, we study existence and uniqueness of blow-up solutions for the problem −Δp(u)+c(x)|∇u|^p−1+F(x,u)=0 in Ω, where 2?p. Under some conditions related to the function F, we give a sufficient condition for existence and nonexistence of nonnegative blow-up solutions. We study also the uniqueness of these solutions.     

On the dependence of the reflection operator on boundary conditions for biharmonic functions

The biharmonic equation arises in areas of continuum mechanics including linear elasticity theory and the Stokes flows, as well as in a radar imaging problem. We discuss the reflection formulas for the biharmonic functions u(x,y)∈R² subject to different boundary conditions on a real-analytic curve in the plane. The obtained formulas, generalizing the celebrated Schwarz symmetry principle for harmonic functions, have different structures. In particular, in the special case of the boundary, Γ₀:={y=0}, reflections are point-to-point when the given on Γ₀ conditions are u=n_∂u=0, u=Δu=0 or n_∂u=n_∂Δu=0, and point to a continuous set when u=n_∂Δu=0 or n_∂u=Δu=0 on Γ₀.     

A uniqueness result for a semilinear elliptic problem: A computer-assisted proof

Starting with the famous article [A. Gidas, W.M. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979) 209-243], many papers have been devoted to the uniqueness question for positive solutions of −Δu=λu+up in Ω, u=0 on ∂Ω, where p>1 and λ ranges between 0 and the first Dirichlet eigenvalue λ₁(Ω) of −Δ. For the case when Ω is a ball, uniqueness could be proved, mainly by ODE techniques. But very little is known when Ω is not a ball, and then only for λ=0. In this article, we prove uniqueness, for all λ∈[0,λ₁(Ω)), in the case Ω=²(0,1) and p=2. This constitutes the first positive answer to the uniqueness question in a domain different from a ball. Our proof makes heavy use of computer assistance: we compute a branch of approximate solutions and prove existence of a true solution branch close to it, using fixed point techniques. By eigenvalue enclosure methods, and an additional analytical argument for λ close to λ₁(Ω), we deduce the non-degeneracy of all solutions along this branch, whence uniqueness follows from the known bifurcation structure of the problem.     

An elementary proof of the blow-up for semilinear wave equation in high space dimensions

This paper concerns the blow-up of solutions to u_tt−Δu=|u|^p in high dimensions for n?4 and 1<p<p₀(n), where p₀(n) is a critical exponent. We proved that the solutions blow up in finite time by estimating the solutions near the wave front using elementary inequalities.     

A note on Keller-Osserman conditions on Carnot groups

This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form Δ_φu≥f(u)l(|∇₀u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δ_φ is the φ-Laplace operator, a natural generalization of the p-Laplace operator which has recently been studied in the context of Carnot groups. We extend to general Carnot groups the results proved in Magliaro et al. (2011) [7] for the Heisenberg group, showing the validity of Liouville-type theorems under a suitable Keller-Osserman condition. In doing so, we also prove a maximum principle for inequality Δ_φu≥f(u)l(|∇₀u|). Finally, we show sharpness of our results for a general φ-Laplacian.