Abstract In this paper we discuss the optimal control of a quasilinear parabolic state equation. Its form is leaned on kind problems arising for example when controlling anisotropic Allen–Cahn equation as model crystal growth. Motivated by application consider result gradient flow an energy functional. The term strongly monotone and obeys certain growth condition lower order non-monotone. discr...

In this paper we establish a geometric theory for abstract quasilinear parabolic equations. In particular, we study existence, uniqueness, and continuous dependence of solutions. Moreover, we give conditions for global existence and establish smoothness properties of solutions. The results are based on maximal regularity estimates in continuous interpolation spaces. An important new ingredient ...

We construct a positive solution to a quasilinear parabolic problem in a bounded spatial domain with the p-Laplacian and a nonsmooth reaction function. We obtain nonuniqueness for zero initial data. Our method is based on suband supersolutions and the weak comparison principle. Using the method of suband supersolutions we construct a positive solution to a quasilinear parabolic problem with the...

In this paper, applying two theorems of Ricceri and Bonanno, we will establish the existence of three weak solutions for a quasilinear elliptic system. Indeed, we will assign a differentiable nonlinear operator to a differential equation system such that the critical points of this operator are weak solutions of the system. In this paper, applying two theorems of R...

In this paper we develop a geometric theory for quasilinear parabolic problems in weighted Lp-spaces. We prove existence and uniqueness of solutions as well as the continuous dependence on the initial data. Moreover, we make use of a regularization effect for quasilinear parabolic equations to study the ω-limit sets and the long-time behaviour of the solutions. These techniques are applied to a...

We are concerned with the hyperbolic Keller-Segel model with quorum sensing, a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities. This is a first order quasilinear equation, its flux depends on space and time via the solution to an elliptic PDE in which the right-hand side is the solution to the hyperbolic equation. This mode...

We deal with the mathematical analysis of the coupling problem in a bounded domain of R, n ≥ 1, between a purely quasilinear first-order hyperbolic equation set on a subdomain and a parabolic one, set on its complementary. We start by providing the definition of a weak solution through an entropy inequality on the whole domain. The uniqueness property relies on a pointwise inequality along the ...