This dissertation describes the space of heteroclinic orbits for a class of semilinear parabolic equations, focusing primarily on the case where the nonlinearity is a second degree polynomial with variable coefficients. Along the way, a new and elementary proof of existence and uniqueness of solutions is given. Heteroclinic orbits are shown to be characterized by a particular functional being f...

Parametric optimal control problems for semilinear parabolic equations are considered. Using recent Lipschitz stability results for solutions of such problems, it is shown that, under standard coercivity conditions, the solutions are Bouligand differentiable (in L, p < ∞) functions of the parameter. The differentials are characterized as the solutions of accessory linear-quadratic problems. A u...

We present a one dimensional semilinear parabolic equation for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. In our example the derivative blows up in the interior of the space interval rather than at the boundary, as in earlier examples. In the case of monotone solutions we show that gradient blow-up occurs at a single...

We study the decay of I (t) = R u(:; t) where u is a nonnegative solution to u t ? u + jruj q = 0 in R n R + with n 1. If 1 q n+2 n+1 then I (t) ! 0 if I (0) < 1 and inf t>0 I (t) > 0 if q > n+2 n+1 .

Introduction Motivations This thesis is devoted to present some new results in the theory of the semilinear approximations to hydrodynamic equations. More specically, the main purpose of this investigation is to prove the convergence of a class of discrete kinetic approximations , the BGK models, to some hyperbolic and hyperbolic-parabolic systems. These BGK models were introduced by Bathnagar,...