Nonlinear Systems of Partial Differential Equations: Applications to Life and Physical Sciences1. Positive solutions for systems of two equations. 1.1. Introduction. 1.2. Strictly positive coexistence for diffusive prey-predator systems. 1.3. Strictly positive coexistence for diffusive competing systems. 1.4. Strictly positive coexistence for diffusive cooperating systems. 1.5. Stability of steady-states as time changes -- 2. Positive solutions for large systems of equations. 2.1. Introduction. 2.2. Synthesizing large (biological) diffusive systems from smaller subsystems. 2.3. Application to epidemics of many interacting infected species. 2.4. Conditions for coexistence in terms of signs of principal eigenvalues of related single equations, mixed boundary data. 2.5. Positive steady-states for large systems by index method. 2.6. Application to reactor dynamics with temperature feedback -- 3. Optimal control for nonlinear systems of partial differential equations. 3.1. Introduction and preliminary results for scalar equations. 3.2. Optimal harvesting-coefficient control of steady-state prey-predator diffusive Volterra-Lotka systems. 3.3. Time-periodic optimal control for competing parabolic systems. 3.4. Optimal control of an initial-boundary value problem for fission reactor systems. 3.5. Optimal boundary control of a parabolic problem -- 4. Persistence, upper and lower estimates, blowup, cross-diffusion and degeneracy. 4.1. Persistence. 4.2. Upper-lower estimates, attractor set, blowup. 4.3. Diffusion, self and cross-diffusion with no-flux boundary condition. 4.4. Degenerate and density-dependent diffusions, non-extinction in highly spatially heterogenous environments -- 5. Traveling waves, systems of waves, invariant manifolds, fluids and plasma. 5.1. Traveling wave solutions for competitive and monotone systems. 5.2. Positive solutions for systems of wave equations and their stabilities. 5.3. Invariant manifolds for coupled Navier-stokes and second order wave equations. 5.4. Existence and global bounds for fluid equations of plasma display technology |
Contents
Positive Solutions for Systems of Two Equations | 1 |
Positive Solutions for Large Systems of Equations | 111 |
values of Related Single Equations Mixed Boundary Data | 143 |
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a₁ assumption asymptotically stable b₁ Banach space bifurcation boundary condition boundary value problem bounded linear operators C₁ Chapter coefficients compact component consider converges Corollary D₁ deduce define denote described in Theorem differential diffusion Dirichlet boundary condition dxdt eigenfunction elliptic equation existence of positive fixed point fixed point index following theorem Fréchet derivative function hypotheses H1 implies inequality interaction invariant manifold K₁ Lemma Leung linear operator lower solution mapping maximum principle monotone Moreover non-negative solution nonlinear obtain optimal control p₁ parameters positive constant positive number principal eigenvalue problem 3.1 proof of Theorem prove respectively satisfies scalar semigroup sequence solution of problem species stability steady-state strictly positive subset sufficiently large sufficiently small Suppose t₁ Theorem 3.1 traveling wave u₁ uniformly upper solution upper-lower solutions v₁ w₁ zero ὃν