We study an abstract nonlinear evolution equation governed by time-dependent operator of subdi erential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary cond...

Global existence, uniqueness and continuous dependence on initial data are established for a quasilinear functional reaction-diiusion equation which arises from a two-dimensional energy balance climate model. Our approach relies heavily on the so-called stability estimates for linear evolution equations of parabolic type (cf. 6]).

Abstract We consider a quasilinear degenerate parabolic equation driven by the orthotropic p -Laplacian. prove that local weak solutions are locally Lipschitz continuous in spatial variable, uniformly time.

In this paper, we consider a model of phosphorus uptake by plant roots, governed by a quasilinear parabolic equation. We first study the well-posedness of the associated Cauchy problem. Then, we consider a shape optimization problem: how to deform the shape of the root in order to increase phosphorus uptake. Finally, we give some numerical results of the shape optimization process.

We establish the existence and uniqueness of the solution to some inner obstacle problems for a coupling of a multidimensional quasilinear firstorder hyperbolic equation set in a region Ωh with a quasilinear parabolic one set in the complementary Ωp = Ω\Ωh. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. Since the interface ∂Ωp ∩ ∂Ωh co...

1.1 Hysteresis Operators 3 1.2 The Duhem Model 13 1.3 Rheological Models of Elasto-plasticity 17 1.4 Discontinuous Hysteresis 35 1.5 Micromagnetism 45 1.6 Models of Hysteresis with Internal Variables 56 1.7 Genesis of Hysteresis and Coupling with PDEs 62 1.8 A Transport Equation with Hysteresis 69 1.9 A Quasilinear Parabolic PDE with Hysteresis 78 1.10 A Quasilinear Hyperbolic PDE with Hysteres...

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...