Boundary estimates for solutions to operators of p-Laplace type with lower order terms

In this paper we study the boundary behavior of solutions to equations of the form

∇⋅A(x,∇u)+B(x,∇u)=0,     

Study of the solutions to a model porous medium equation with variable exponent of nonlinearity

The authors of this paper study the Dirichlet problem of the following equation

ut−div(|u|^ν(x,t)∇u)=f−|u|^p(x,t)−1u.     

Existence of a unique solution to a quasilinear elliptic equation

The purpose of this paper is to prove the existence of a unique classical solution u(x) to the quasilinear elliptic equation −∇⋅(a(u)∇u)+v⋅∇u=f, where u(x₀)=u₀ at x₀∈Ω and where n⋅∇u=g on the boundary ∂Ω. We prove that if the functions a, f, v, g satisfy certain conditions, then a unique classical solution u(x) exists. Applications include stationary heat/diffusion problems with convection and with a source/sink, where the value of the solution is known at a spatial location x₀∈Ω, and where n⋅∇u is known on the boundary.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

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)     

On the complete group classification of the reaction-diffusion equation with a delay

The reaction-diffusion delay differential equation

ut(x,t)−uxx(x,t)=g(x,u(x,t),u(x,t−τ))     

Li-Yau type estimates for a nonlinear parabolic equation on complete manifolds

Let (M,g) be a complete noncompact Riemannian manifold with the m-dimensional Bakry-Émery Ricci curvature bounded below. In this paper, we give a local Li-Yau type gradient estimate for the positive solutions to a general nonlinear parabolic equation

ut=Δu−∇?⋅∇u−aulogu−qu     

A Riemann-Hilbert Approach to the Chen-Lee-Liu Equation on the Half Line

In this paper, the Fokas unified method is used to analyze the initial-boundary value for the Chen- Lee-Liu equation

$i{\partial _t}u + {\partial_{xx}u - i |u{|^2}{\partial _x}u = 0}$

on the half line (?∞, 0] with decaying initial value. Assuming that the solution u(x, t) exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter λ. The jump matrix has explicit (x, t) dependence and is given in terms of the spectral functions {a(λ), b(λ)} and {A(λ), B(λ)}, which are obtained from the initial data u₀(x) = u(x, 0) and the boundary data g₀(t) = u(0, t), g₁(t) = u_x(0, t), respectively. The spectral functions are not independent, but satisfy a so-called global relation.

     

An inverse problem of identifying the coefficient in a nonlinear parabolic equation

This paper deals with the determination of a pair (p,u) in the nonlinear parabolic equation

u_t−u_xx+p(x)f(u)=0,     

On a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws

In this paper we study the initial problem for a stochastic nonlinear equation arising from 1D integro-differential scalar conservation laws. The equation is driven by Lévy space-time white noise in the following form:

(t_∂−A)u+x_∂q(u)=f(u)+g(u)Ft,x     

Weighted doubling properties and unique continuation theorems for the degenerate Schrödinger equations with singular potentials

Let u be the weak solution to the degenerate Schrödinger equation with singular coefficients in Lipschitz domain as following

−div(w(x)A(x)∇u(x))+V(x)u(x)w(x)=0,     

Existence, uniqueness and stability of travelling waves in a discrete reaction-diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:

u_t(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),     

Gaussian bounds for degenerate parabolic equations

Let A be a real symmetric, degenerate elliptic matrix whose degeneracy is controlled by a weight w in the A₂ or QC class. We show that there is a heat kernel Wt(x,y) associated to the parabolic equation wut=divA∇u, and Wt satisfies classic Gaussian bounds:

     

Spectral stability of shock waves associated with not genuinely nonlinear modes

We study viscous shock waves that are associated with a simple mode (λ,r)$(\lambda ,r)$ of a system _ut+f_(u)x=_uxx$u_t+f{(u)}_x=u_{xx}$ of conservation laws and that connect states on either side of an ‘inflection’ hypersurface Σ   in state space at whose points r⋅∇λ=0$r\cdot \mathrm{\nabla }\lambda =0$ and (r⋅∇)²λ≠0${(r\cdot \mathrm{\nabla })}^2\lambda \ne 0$. Such loss of genuine nonlinearity, the simplest example of which is the cubic scalar conservation law _ut+_(u3)x=_uxx$u_t+{(u^3)}_x=u_{xx}$, occurs in many physical systems. We show that such shock waves are spectrally stable if their amplitude is sufficiently small. The proof is based on a direct analysis of the eigenvalue problem by means of geometric singular perturbation theory. Well-chosen rescalings are crucial for resolving degeneracies. By results of Zumbrun the spectral stability shown here implies nonlinear stability of these shock waves.     

Existence of solutions for a cantilever beam problem

We are concerned with the fourth-order nonuniform cantilever beam problem

(I(x)WΔ∇(x)⁾Δ∇=f(x,W(x)),     

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.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Global existence and blowup solutions for quasilinear parabolic equations

The authors discuss the quasilinear parabolic equation ut=∇⋅(g(u)∇u)+h(u,∇u)+f(u) with u_|∂Ω=0, u(x,0)=?(x). If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.     

Sobolev regularization of a class of forward–backward parabolic equations

We address existence and asymptotic behaviour for large time of Young measure solutions   of the Dirichlet initial–boundary value problem for the equation _ut=∇⋅[φ(∇u)]$u_t=\mathrm{\nabla }\cdot [\phi (\mathrm{\nabla }u)]$, where the function φ need not satisfy monotonicity conditions. Under suitable growth conditions on φ  , these solutions are obtained by a “vanishing viscosity” method from solutions of the corresponding problem for the regularized equation _ut=∇⋅[φ(∇u)]+?Δ_ut$u_t=\mathrm{\nabla }\cdot [\phi (\mathrm{\nabla }u)]+?\mathrm{\Delta }{u}_t$. The asymptotic behaviour as t→∞$t\to \infty$ of Young measure solutions of the original problem is studied by ω-limit set techniques, relying on the tightness   of sequences of time translates of the limiting Young measure. When N=1$N=1$ this measure is characterized as a linear combination of Dirac measures with support on the branches of the graph of φ.     

Gevrey micro-regularity for solutions to first order nonlinear PDE

Let (x,t)∈Rm×R and u∈C²(Rm×R). We study the Gevrey micro-regularity of solutions u of the nonlinear equation

ut=f(x,t,u,ux),