Trajectories joining critical points of nonlinear parabolic and hyperbolic partial differential equations

Consider the following nonlinear Dirichlet boundary value problems:

$\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+f(u(t,x)),}\text{t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.1)

$\text{D}{}_{\mathrm{t}}\text{D}{}_{\mathrm{t}}\text{u(t,x)=Lu(t,x)+}\text{α}\text{L(D}{}_{\mathrm{t}}\text{u(t,x))+f(u(t,x)),}\text{α}\text{>0,t?0,x∈Ω}\text{u(t,x)=0,}\text{t?0,x∈?Ω}$

(1.2)

$\text{Lu(x)+f(u(x)),}\text{x∈Ω}\text{u(x)=0,}\text{x∈?Ω}$

. (1.3) In all of these equations, f: R → R is a locally Lipschitzian asymptotically linear function with positive asymptotic slope, f(0) = 0, and L is a self-adjoint, negativedefinite and strongly elliptic second-order differential operator on a smooth domain Ω in Rⁿ. The solutions of (1.1) and (1.2) generate semiflows which are not pointdissipative and whose equilibria are determined by solutions of (1.3). In this paper, using an extension (due to the present author) of Conley's Morse index theory to noncompact spaces, we prove not only the existence of positive solutions of (1.3) (a result shown earlier by Peitgen and Schmitt using different methods), but also show the existence of (nonconstant) heteroclinic orbits of (1.1) and (1.2) joining two sets of equilibria.