Traveling wave solutions of nonlinear conservation laws arising from image processing and from chemotaxis

In this thesis, we study nonlinear partial differential equations arising from image processing and cheomotaxis. We analyze mathematical models in conservative form from the perspective of traveling wave solutions. We show the existence and the stability of traveling wave solutions in the models, which helps to understand the behaviors of solutions in the models. The thesis largely consists of two parts: (1) We develop stability analysis for a traveling wave solution of a nonlinear conservation law arising from image processing. To be specific, we prove that if the initial perturbation between a solution and a traveling wave solution to the problem is small, the solution converges to the traveling wave solution.To show this, we construct a weight function in establishing energy estimates to overcome difficulties caused by the absence of the convexity of a flux of the conservation law. (2) We develop dynamical systems theory to study traveling wave solutions in a chemotaxis model. A traveling wave solution to the model in a partial differential equation is a heteroclinic/homoclinic orbit to the model in an ordinary differential equation. Thus, we investigate the existence and non-existence of a heteroclinic/homoclinic orbit in certain ranges of parameters in the model by applying dynamical systems theory.