Wave-Like Solutions for a Class of Parabolic Models

A reduction aproach is developed for determining exact solutions of anonlinear second order parabolic PDE. The method in point makes acomplementary use of the leading ideas of the theory of quasilinearhyperbolic systems of first order endowed by differential constraintsand of the techniques providing multiple wave-like solutions ofnonlinear PDEs. The searched solutions exhibit a inherent wave featuresand they are obtained by solving a consistent overdetermined system ofPDEs. Remarkably, in the process it is possible to define nonlinearmodel equations which allow special classes of initial or boundary valueproblems to be solved in a closed form. Within the present reductionapproach exact solutions and model material response functions areobtained for an equation of widespread application in many fields ofinterest.     

A Geometrical Approach to the Study¶of Unbounded Solutions¶of Quasilinear Parabolic Equations

In this article, we are interested in the existence and uniqueness of solutions for quasilinear parabolic equations set in the whole space ? ^N . We consider, in particular, cases when there is no restriction on the growth or the behavior of these solutions at infinity. Our model equation is the mean-curvature equation for graphs for which Ecker and Huisken have shown the existence of smooth solutions for any locally Lipschitz continuous initial data. We use a geometrical approach which consists in seeing the evolution of the graph of a solution as a geometric motion which is then studied by the so-called “level-set approach”. After determining the right class of quasilinear parabolic PDEs which can be taken into account by this approach, we show how the uniqueness for the original PDE is related to “fattening phenomena” in the level-set approach. Existence of solutions is proved using a local L ^∞ bound obtained by using in an essential way the level-set approach. Finally we apply these results to convex initial data and prove existence and comparison results in full generality, i.e., without restriction on their growth at infinity.     

Global Continua of Positive Solutions for some Quasilinear Parabolic Equation with a Nonlocal Initial Condition

This paper is concerned with a quasilinear parabolic equation including a nonlinear nonlocal initial condition. The problem arises as equilibrium equation in population dynamics with nonlinear diffusion. We make use of global bifurcation theory to prove existence of an unbounded continuum of positive solutions.     

Quasivariational Solutions for First Order Quasilinear Equations with Gradient Constraint

We prove the existence of solutions for a quasi-variational inequality of evolution with a first order quasilinear operator and a variable convex set which is characterized by a constraint on the absolute value of the gradient that depends on the solution itself. The only required assumption on the nonlinearity of this constraint is its continuity and positivity. The method relies on an appropriate parabolic regularization and suitable a priori estimates. We also obtain the existence of stationary solutions by studying the asymptotic behaviour in time. In the variational case, corresponding to a constraint independent of the solution, we also give uniqueness results.     

Regularity Results for Solutions of a Class of Hamilton-Jacobi Equations

The regularity of the gradient of viscosity solutions of first‐order Hamilton‐Jacobi equations is studied under a strict convexity assumption on H(t,x,⋅). Estimates on the discontinuity set of Du are derived. Such estimates imply that solutions of the above problem are smooth in the complement of a closed ℋ ⁿ ‐rectifiable set. In particular, it follows that Du belongs to the classSBV, i.e., D ² u$ is a measure with no Cantor part. (Accepted February 12, 1996)     

On the First Boundary Value Problem for Quasilinear Parabolic Equations with Two Independent Variables

This paper is concerned with the global solvability of the first initial boundary value problem for the quasilinear parabolic equations with two independent variables: a(t,x,u,u_xINF>)u_xxm u_t=f(t,x,u,u_xINF>). We investigate the case when the growth of [(|f(t,x,u,p)|)/(a(t,x,u,p))]{{|f(t,x,u,p)|}\over {a(t,x,u,p)}} with respect to p is faster than p² when |p|M X. Conditions which guarantee the global classical solvability of the problem are formulated.     

Estimates of Solutions and Asymptotic Symmetry for Parabolic Equations on Bounded Domains

We consider fully nonlinear parabolic equations on bounded domains under Dirichlet boundary conditions. Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared with previous results of this type, we do not assume certain crucial hypotheses, such as uniform (with respect to time) positivity of the solution or regularity of the nonlinearity in time. Our method is based on estimates of solutions of linear parabolic problems, in particular on a theorem on asymptotic positivity of such solutions.     

The Extinction Behavior of the Solutions for a Class of Reaction-Diffusion Equations

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On Asymptotic Properties of Continuously Differentiable Solutions of Quasilinear Functional Differential Equations

We establish new properties of C ¹ [τ(1), + ∞)-solutions of the quasilinear functional differential equation

Pseudo Almost Periodic Solutions to a Class of Semilinear Differential Equations

This paper is concerned with the existence and uniqueness of pseudo almost periodic solutions to a class of semilinear differential equations involving the algebraic sum of two (possibly noncommuting) densely defined closed linear operators acting on a Hilbert space. Sufficient conditions for the existence and uniqueness of pseudo almost periodic solutions to those semilinear equations are obtained. An erratum to this article is available at .     

Backward Uniqueness for Parabolic Equations

It is shown that a function u satisfying |_t+u|M(|u|+|u|), |u(x, t)|Me^M|x|2 in (ⁿ \ (B_R) × [0, T] and u(x, 0) = 0 for xⁿ \ B_R must vanish identically in ⁿ \ B_R×[0, T].     

Quasilinear Equations for Viscoelasticity of Strain-Rate Type

We consider the quasilinear m ×  m system of partial differential equations that governs the motion of a viscoelastic material of strain-rate type on a bounded domain in . The dependence of the stress on both the strain and strain-rate is nonlinear, and our hypotheses allow for a potential energy which is a nonconvex function of the strain. The critical hypothesis is that the dependence of the stress function on the strain rate is uniformly strictly monotone (in the sense of Minty and Browder). We prove the existence and uniqueness of weak solutions to a natural initial-boundary value problem for a large class of constitutive functions. We then treat the question of H ²-regularity of solutions and show that, while regularity in the initial data is preserved, solutions do not, in general, become more regular than their initial data. This generalizes a result for the semilinear case due to Rybka.     

New Explicit and Exact Travelling Wave Solutions for a Class of Nonlinear Evolution Equations

ＩｎｔｒｏｄｕｃｔｉｏｎＷｉｔｈｔｈｅｒａｐｉｄｄｅｖｅｌｏｐｍｅｎｔｏｆｓｃｉｅｎｃｅａｎｄｔｅｃｈｎｏｌｏｇｙ ,ｔｈｅｓｔｕｄｙｋｅｒｎｅｌｏｆｍｏｄｅｒｎｓｃｉｅｎｃｅｉｓｃｈａｎｇｅｄｆｒｏｍｌｉｎｅａｒｔｏｎｏｎｌｉｎｅａｒｓｔｅｐｂｙｓｔｅｐ .Ｍａｎｙｎｏｎｌｉｎｅａｒｓｃｉｅｎｃｅｐｒｏｂｌｅｍｓｃａｎｓｉｍｐｌｙａｎｄｅｘａｃｔｌｙｂｅｄｅｓｃｒｉｂｅｄｂｙｕｓｉｎｇｔｈｅｍａｔｈｅｍａｔｉｃａｌｍｏｄｅｌｏｆｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎ .Ｕｐｔｏｎｏｗ ,ｍａｎｙｉｍｐｏｒ…     

Absolute Exponential Stability of Solutions of Linear Parabolic Differential Equations with Delay

We find necessary and sufficient conditions for the absolute exponential stability of solutions of linear parabolic differential equations with delay in a pair of norms.     

Exponential Dichotomy for a Nonautonomous System of Parabolic Equations

In this paper we study the existence and roughness of exponential dichotomy (ED) of a non-autonomous system of parabolic equations with Neumann boundary conditions. In order to do that, we first set the problem in the Linear Skew-Product Semiflow (LSPS) framework. Then we prove that the ED is not destroyed by small perturbation (roughness). Next, we compute the dynamical spectrum for this LSPS. Finally, under some conditions we prove that zero does not belong to the dynamical spectrum corresponding to this LSPS. i.e., the system has ED (existence).     

Laplace Type Invariants for Parabolic Equations

The Laplace invariants pertain to linear hyperbolic differentialequations with two independent variables. They were discovered byLaplace in 1773 and used in his integration theory of hyperbolicequations. Cotton extended the Laplace invariants to ellipticequations in 1900. Cotton's invariants can be obtained from the Laplaceinvariants merely by the complex change of variables relating theelliptic and hyperbolic equations.To the best of my knowledge, the invariants for parabolic equations werenot found thus far. The purpose of this paper is to fill this gap byconsidering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found forparabolic equations.     

Periodic Step-Like Contrast Structures for a Quasilinear Parabolic Equation with Small Diffusion Coefficient

Using the method of boundary functions, for a quasilinear parabolic equation with small diffusion coefficient we construct an asymptotic expansion of a periodic solution with internal transition layer. Sufficient conditions for the existence of this solution are obtained. __________ Translated from Neliniini Kolyvannya, Vol. 8, No. 3, pp. 329–350, July–September, 2005.     

Application of Topological Methods to Quasilinear Parabolic Boundary-Value Problems

We apply a topological approach to the investigation of quasilinear parabolic boundary-value problems. The class of problems under investigation is reduced to an operator equation with an operator satisfying condition (S)₊. We establish theorems on solvability and give an example of the application of the approach indicated to the case of second-order parabolic equations.     

Non-Existence of Global Solutions For a Quasilinear Benney System

Benney introduced in 1977 (cf. Stud Appl Math 56:81–94, 1977) a general strategy for deriving systems of nonlinear PDEs describing the interaction between long and short waves. In Dias et al. (CR Acad Sci Paris I 344:493–496, 2007) we have studied the local existence and unicity of solutions to a quasilinear version of these systems. In the present paper we prove that in some important cases global strong solutions do not exist.     

A Strongly Degenerate Quasilinear Equation: the Parabolic Case

We prove the existence and uniqueness of entropy solutions of the Neumann problem for the quasilinear parabolic equation u_t=÷ a(u, Du), where a(z,)=f(z,), and f is a convex function of with linear growth as ||||, satisfying other additional assumptions. In particular, this class includes the case where f(z,)=(z)(), >0, and is a convex function with linear growth as ||||.