The Study Of Dissipative Hyperbolic Mean Curvature Flow

The successful application of elliptic and parabolic partial differential equations in differential geometry and physics has inspired scholars to study hyperbolic differential equation theory.Hyperbolic mean curvature flow has been widely used in the fields of crystal evolution and biomedicine,and achieved many great results.On this basis,this paper studied the Cauchy problem and evolution of plane curves for dissipative hyperbolic mean curvature flow.Chapter 1 introduced the research background of dissipative hyperbolic mean curvature flow and the main research achievements of this paper.Chapter 2 investigated the initial value problems of the dissipative hyperbolic mean curvature flow.A hyperbolic Monge-Ampère equation is derived by means of support functions for a convex curve,and this equation could be reduced to the first order quasilinear systems in Riemann invariants.Using the theory of the local solutions of Cauchy problems for quasilinear hyperbolic systems,we discussed lower bounds on life-span of classical solutions to Cauchy problems for dissipative hyperbolic mean curvature flow.Chapter 3 investigated the evolution of convex plane curves for dissipative hyperbolic mean curvature flow.We investigated some exact solutions and gave an example to further explain the dissipative hyperbolic mean curvature flow.Particularly,we gave some propositions and proved the result.If the minimum of initial velocity greater than or equal to zero,the flow will converge to a point or become a curve which has the discontinuous curvature in a finite time.If the maximum of initial velocity lesser than zero,the flow will expand first and then converge to a point or become a curve which has the discontinuous curvature in a finite time.In addition,I studied the application of tanh-function methods.In appendix,I applied the modified extended tanh-function method to mK(m,n)equation and obtained the exact travelling wave solutions.And the travelling wave solutions are expressed by thehyperbolic tangent and trigonometric functions.In addition,we gave several partial figures of some special exact travelling wave solutions.