Gregorio Díaz

Some properties of solutions of degenerate second order P.D.E. in non-divergence form

In this paper we study some properties of solution of F(x,u,D2u)=f. The appearance of a null set is derived from an assumption of degeneracy at the origin, involving the dependence of F on the zeroth and second derivative terms. The non appearance of a null set is obtained by the strong maximum principle of Hopf when a strong non degeneracy assumption holds. Interior properties of solutions of F(x,u,D2u)=f on RN are also obtained under another assumption of degeneracy at infinity.

Unbounded solutions of one dimensional quasilinear elliptic equations

Unbounded nonnegative solutions of are obtained, provided a necessary and sufficient strong interior condition on the structure of the equation, as well as the boundedness of ]a, b[. With no loss of generality, the study is developed in the class of solutions of (e) satisfying the boundary values . Local comparison and other interior properties are also deduced.

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeut−R¦▽u¦m+λuq=0 in ℝn× ℝ+, u(·,0+)=uo(·) in ℝN, whereR > 0, 0 <q < 1,m ≥ 1, anduo is a positive uniformly continuous function verifying −R¦▽uo¦m+λu0q⩾ 0 in ℝN. We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t∞(·) defined byu(x, t) > 0 if 0<t<t∞(x) andu(x, t)=0 ift ≥t∞(x). Regularity, extinction rate, and asymptotic behavior of t∞(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(uo(x − ξt))1−q −λ(1−q)t]+)1/(1−q): ¦ξ¦≤R}, (x, t)εℝ+N+1.

Blow-up time involved with perturbed Hamilton-Jacobi equations

In this paper we consider the evolution of positive bounded uniformly continuous data u 0 by perturbed equations like Under general assumptions on u 0 , existence, uniqueness and regularity of the evolution u in the set are studied, where the blow-up function is given by . The exact blow-up rate of u is obtained. Uniqueness, regularity, decay at infinity of the function , as well a s a representation formula for the case m = 1, are also proved.