On similarity solutions for boundary layer flows with prescribed heat flux

This paper is concerned with existence, uniqueness and behaviour of the solutions of the autonomous third‐order non‐linear differential equation f?+(m+2)f f″?(2m+1)f′²=0 on ?⁺ with the boundary conditions f(0)=?γ, f′(∞)=0 and f″(0)=?1. This problem arises when looking for similarity solutions for boundary layer flows with prescribed heat flux. To study solutions we use some direct approach as well as blowing‐up co‐ordinates to obtain a plane dynamical system. Copyright © 2004 John Wiley & Sons, Ltd.     

Nonlinear degenerate parabolic equations for Baouendi–Grushin operators

In this paper, we shall investigate the nonexistence of positive solutions for the following nonlinear parabolic partial differential equations: and Here, Ω is a Carnot–Carathéodory metric ball in R ^N and V ∈ L ¹_loc(Ω). The critical exponents m * and p * are found, and the nonexistence results are proved for m * ≤ m < 1 and p * ≤ p < 2. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On non‐Newtonian incompressible fluids with phase transitions

A modified model for a binary fluid is analysed mathematically. The governing equations of the motion consists of a Cahn–Hilliard equation coupled with a system describing a class of non‐Newtonian incompressible fluid with p‐structure. The existence of weak solutions for the evolution problems is shown for the space dimension d=2 with p? 2 and for d=3 with p? 11/5. The existence of measure‐valued solutions is obtained for d=3 in the case 2? p< 11/5. Similar existence results are obtained for the case of nondifferentiable free energy, corresponding to the density constraint |ψ| ? 1. We also give regularity and uniqueness results for the solutions and characterize stable stationary solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

Existence and uniqueness of minimizers for L2-constrained problems related to fractional Kirchhoff equation

In this paper, we discuss the existence and uniqueness of solutions of the constrained variational problem with respect to the fractional Kirchhoff equation. For the exponent p<p_*(s,N), a complete classification with respect to p for the existence of solutions of the fractional Kirchhoff functional on the L²-normalized manifold was given. Furthermore, all these solutions are unique up to translations, and our methods depend only on some simple energy estimates.     

Existence and uniqueness of weak solutions for some singular evolutionary system in image denoising

In this paper, we prove the existence and uniqueness of weak solutions for some singular evolutionary system, which is deduced from one of the Chambolle–Lions denoising models. The main method we used is p‐Laplace regularization. Copyright © 2013 John Wiley & Sons, Ltd.     

Blowup of solutions for a class of non‐linear evolution equations with non‐linear damping and source terms

We consider the blowup of solutions of the initial boundary value problem for a class of non‐linear evolution equations with non‐linear damping and source terms. By using the energy compensation method, we prove that when p>max{m, α}, where m, α and p are non‐negative real numbers and m+1, α+1, p+1 are, respectively, the growth orders of the non‐linear strain terms, damping term and source term, under the appropriate conditions, any weak solution of the above‐mentioned problem blows up in finite time. Comparison of the results with the previous ones shows that there exist some clear condition boundaries similar to thresholds among the growth orders of the non‐linear terms, the states of the initial energy and the existence and non‐existence of global weak solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Reconstructing the past from imprecise knowledge of the present: Effective non‐uniqueness in solving parabolic equations backward in time

Identifying sources of ground water pollution and deblurring astronomical galaxy images are two important applications generating growing interest in the numerical computation of parabolic equations backward in time. However, while backward uniqueness typically prevails in parabolic equations, the precise data needed for the existence of a particular backward solution is seldom available. This paper discusses previously unexplored non‐uniqueness issues, originating from trying to reconstruct a particular solution from imprecise data. Explicit 1D examples of linear and nonlinear parabolic equations are presented, in which there is strong computational evidence for the existence of distinct solutions w^red(x,t) and w^green(x,t), on 0 ≤ t ≤ 1. These solutions have the property that the traces w^red(x,1) and w^green(x,1) at time t = 1 are close enough to be visually indistinguishable, while the corresponding initial values w^red(x,0) and w^green(x,0) are vastly different, well‐behaved, physically plausible functions, with comparable L² norms. This implies effective non‐uniqueness in the recovery of w^red(x,0) from approximate data for w^red(x,1). In all these examples, the Van Cittert iterative procedure is used as a tool to discover unsuspected, valid, additional solutions w^green(x,0). This methodology can generate numerous other examples and indicates that multidimensional problems are likely to be a rich source of striking non‐uniqueness phenomena. Published 2012. This article is a US Government work and is in the public domain in the USA.     

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.     

Lp Solutions for Multidimensional BDSDEs with Locally Weak Monotonicity Coefficients

In this paper, the authors establish the existence and uniqueness theorem of L^p (1 < p ≤ 2) solutions for multidimensional backward doubly stochastic differential equations (BDSDEs for short) under the p-order globally (locally) weak monotonicity conditions. Comparison theorem of L^p solutions for one-dimensional BDSDEs is also proved.These conclusions unify and generalize some known results.