Global Solutions to Elliptic and Parabolic $${\Phi^4}$$ Φ 4 Models in Euclidean Space

Parallel Transport Frame in 4 -dimensional Euclidean Space

In this work, we give parallel transport frame of a curve and we introduce the relations between the frame and Frenet frame of the curve in 4-dimensional Euclidean space. The relation which is well known in Euclidean 3-space is generalized for the …rst time in 4-dimensional Euclidean space. Then we obtain the condition for spherical curves using the parallel transport frame of them. The conditi...

Asymptotic analysis of solutions to elliptic and parabolic problems

In the thesis we consider two types of problems. In Paper 1, we study small solutions to a time-independent nonlinear elliptic partial differential equation of Emden-Fowler type in a semi-infinite cylinder. The asymptotic behaviour of these solutions at infinity is determined. First, the equation under the Neumann boundary condition is studied. We show that any solution small enough either vani...

Global existence of solutions to a parabolic-elliptic chemotaxis system with critical degenerate di ffusion

Viscosity solutions for elliptic-parabolic problems

We study an elliptic-parabolic problem appearing in the theory of partially saturated flows in the framework of viscosity solutions. This is part of current investigation to understand the theory of viscosity solutions for PDE problems involving free boundaries. We prove that the problem is well posed in the viscosity setting and compare the results with the weak theory. Dirichlet or Neumann bo...

Uniqueness of Entropy Solutions of Nonlinear Elliptic-Parabolic-Hyperbolic Problems in One Dimension Space

We consider a class of elliptic-parabolic-hyperbolic degenerate equations of the form b(u)t−a(u, φ(u)x)x = f with homogeneous Dirichlet conditions and initial conditions. In this paper we prove an L-contraction principle and the uniqueness of entropy solutions under rather general assumptions on the data.