Showing papers in "Electronic Journal of Differential Equations in 1998"

Optimizing chemotherapy in an hiv model

Abstract: We examine an ordinary dierential system modeling the interaction of the HIV virus and the immune system of the human body. The optimal control representsapercentageeect thechemotherapyhasontheinteractionoftheCD4 + T cells withthevirus. WemaximizethebenetbasedontheTcell countandminimize the systemic cost based on the percentage of chemotherapy given. Existence of an optimal control is proven, and the optimal control is uniquely characterized in terms of the solution of the optimality system, which is the state system coupled with the adjoint system. In addition, numerical examples are given for illustration.

A minmax principle, index of the critical point, and existence of sign-changing solutions to elliptic boundary value problems

Abstract: In this article we apply the minmax principle we developed in [6] to obtain sign-changing solutions for superlinear and asymptotically linear Dirichlet problems. We prove that, when isolated, the local degree of any solution given by this minmax principle is +1. By combining the results of [6] with the degree-theoretic results of Castro and Cossio in [5], in the case where the nonlinearity is asymptotically linear, we provide sucient conditions for: i) the existence of at least four solutions (one of which changes sign exactly once), ii) the existence of at leastve solutions (two ofwhichchangesign), andiii)theexistenceofpreciselytwosign-changing solutions. For a superlinear problem in thin annuli we prove: i) the existence of a non-radial sign-changing solution when the annulus is suciently thin, andii)theexistenceofarbitrarilymanysign-changingnon-radialsolutions when, in addition, the annulus is two dimensional. The reader is referred to [7] where the existence of non-radial signchanging solutions is established when the underlying region is a ball.

Adjoint and Self-adjoint Differential Operators on Graphs

Abstract: A differential operator on a directed graph with weighted edges is characterized as a system of ordinary differential operators. A class of local operators is introduced to clarify which operators should be considered as defined on the graph. When the edge lengths have a positive lower bound, all local self-adjoint extensions of the minimal symmetric operator may be classified by boundary conditions at the vertices.

Existence of continuous and singular ground states for semilinear elliptic systems

Abstract: Westudyexistenceresultsofacurveofcontinuousandsingularground states for the system u=(jxj)f(v) v=(jxj)g(u); wherex2 R N nf0g, the functionsf andg are increasing Lipschitz continuousfunctions in R,andandare nonnegative continuousfunctions in R + . We also study general systems of the form u(x)+V(jxj)u+a(jxj)v p =0 v(x)+V(jxj)v+b(jxj)u q =0:

On a mixed problem for a linear coupled system with variable coefficients.

Abstract: We prove existence, uniqueness and exponential decay of solutions to the mixed problem u(x, t)− μ(t)∆u(x, t) + ∑n i=1 ∂θ ∂xi (x, t) = 0 , θ(x, t)−∆θ(x, t) + ∑n i=1 ∂u′ ∂xi (x, t) = 0 , with a suitable boundary damping, and a positive real-valued function μ.

Existence and regularity results for the gradient flow for p-harmonic maps

Abstract: We establish existence and regularity for a solution of the evolution problem associated to p-harmonic maps if the target manifold has a nonpositive sectional curvature.