Hans Engler

RELATIONS BETWEEN TRAVELLING WAVE SOLUTIONS OF QUASILINEAR PARABOLIC EQUATIONS

On donne une transformation entre les solutions onde de propagation de v t =v xx +f(v) et u t =(D(u)u x ) x +g(u)

Weak solutions of a class of quasilinear hyperbolic integro-differential equations describing viscoelastic materials

Global weak solutions of scalar second-order quasilinear hyperbolic integro-differential equations with singular kernels are constructed. Perturbations of rest states are shown to propagate with finite speed, smoothing effects of the solution operator are exhibited, and conditions for the asymptotic stability of rest states are given. The equations arise in viscoelasticity.

Global regular solutions for the dynamic antiplane shear problem in nonlinear viscoelasticity

Here A is the two-dimensional Laplacian, g is a scalar function, and Uo, u~, f are given functions. Equation (1) occurs in the description of antiplane shear motions of certain viscoelastic solids; for background material, the reader is referred to [-20]. Let y(~, t) denote the position of a particle with reference position ~ at time t, and consider an infinite column ~2 • R consisting of a homogeneous isotropic incompressible solid with the semilinear constitutive equation

Gradient estimates for solutions of parabolic equations and systems

1. STATEMENT OF THE PROBLEM Let 52 c R”, n > 2, be a bounded domain with smooth boundary r= &2. In this note we study scalar quasilinear parabolic differential equations u, = div,Y(g( IVul’) VU) +f(u) = 0, (1) u,-A,u+f(u)=O (2) and systems u, DA,u + f(u) = 0 (3) on Q x (0, co), with initial conditions u( ., 0) = u0 resp. u( ., 0) = u0 and homogeneous Dirichlet boundary conditions u,i-X(O,CO)‘O for (1) and (2) resp. (4) u1r.x (0.00) = 0 for (3). (5)

On the Speed of Spread for Fractional Reaction-Diffusion Equations

The fractional reaction diffusion equation ∂tu Au g u is discussed, where A is a fractional differential operator on R of order α ∈ 0, 2 , the C1 function g vanishes at ζ 0 and ζ 1, and either g ≥ 0 on 0, 1 or g 1, or if g is merely positive on a sufficiently large interval near ζ 1 in the case α < 1. On the other hand, it shown that solutions spread with finite speed if g ′ 0 < 0. The proofs use comparison arguments and a suitable family of travelling wave solutions.

On the dynamic shear flow problem for viscoelastic liquids

nitial-boundary value problems for a third order nonlinear integro-differential equation describing dynamic simple shear flow for viscoelastic liquids are studied on bounded one-dimensional spatial domains. Local and global existence results for arbitrary forces and initial data are given under suitable assumptions on the constitutive relations. Conditions on the forces and on the constitutive equations are formulated that imply that solutions of the equations tend to a rest state, and the convergence rates are estimated in terms of the force decay and of dissipation rates that can be derived from the constitutive equations.

Asymptotic Self-Similarity for Solutions of Partial Integro-Differential Equations

The question is studied whether weak solutions of linear partial integrodifferential equations approach a constant spatial profile after rescaling, as time goes to infinity. The possible limits and corresponding scaling functions are identified and are shown to actually occur. The limiting equations are fractional diffusion equations which are known to have self-similar fundamental solutions. For an important special case, is is shown that the asymptotic profile is Gaussian and convergence holds in L, that is, solutions behave like fundamental solutions of the heat equation to leading order. Systems of integrodifferential equations occurring in viscoelasticity are also discussed, and their solutions are shown to behave like fundamental solutions of a related Stokes system. The main assumption is that the integral kernel in the equation is regularly varying in the sense of Karamata. 45K05; 35B40

Contractive Properties for the Heat Equation in Sobolev Spaces

Abstract Consider the heat equation ∂ r u − Δ x u = 0 in a cylinder Ω × [0, T ] ⊂ R n +1 smooth lateral boundary under zero Neumann or Dirichlet conditions. Geometric conditions for Ω are given that guarantee that for a given P , ‖▽ x u (·, t )‖ L p will be non-increasing for any solution. Decay rates are also given. For arbitrary Ω and p , it is shown how to construct an equivalent L p -norm, such that ▽ x (·, t ) is non-increasing in this norm.

Stabilization of Solutions for a Class of Parabolic Integro-Differential Equations.

Abstract : Integro-differential equations arise in the description of feed-back control systems, where the control variables are derived from filtered observations of the state or where the control mechanism possesses inertia. The author studies a model equation for a distributed control system (e.g., the state varies over some space-like domain) which contains also some diffusion effects and give conditions under which the state will tend to some limit, as time goes to infinity, regardless of the initial situation. The limit is shown to satisfy an elliptic differential equation. Convergence rates are also given; these show the slowing-down effect of a slow control mechanism on the convergence of the state variable. The problem under study can also be viewed as a natural extension of a type of reaction-diffusion equation that has received wide attention in the literature. (Author)

A Version of the Chain Rule and Integrodifferential Equations in Hilbert Spaces

For Hilbert-space valued functions u, subdifferentials