Functional differential inequalities of parabolic type

Abstract

We prove a theorem wich generalizes a result of J. Szarski (see [4];Theor. 2)concerning weak inequalities fora diagonal system of second order differential functional inequalities of hetype $u_{t}ⁱ≤ fⁱ(t,x,u(t,x), u_{z}ⁱ(t,x), u_{x,x}ⁱ(t,x), u(t,·))$ $i = 1,...,m$, assuming that $fⁱ$ is parabolic with respect to u for any $i=1,...,m$. After introducing the definition of left parabolic (or right parabolic) function with respect to another one, we obtain the over mentioned generalization (Theor. 2.2) as a consequence of a theorem about strong inequalities (Theor. 2.1) which is a generalization of Theor.1 of [4] in the case of left parabolic (or right parabolic) functions. These generalizations have been suggested by the following example in wich we have the assertion of Theor. 2 of [4] even if hypotheses of the theorem are not al1 verified. Consider the function f defined as (l) $f(t,x,u,q,r,z) = (x_{1}² + x_{2}² - 1)sgn r_{11}$ for $(t,x) ∈ d = ]0,T[?{x = (x₁, x₂) ∈ {R}² : x_{1}² + x_{2}² < 1}, u in {R}, q ∈ {R}², r = (r_{ij})_{1 ≤ ij ≤ 2}$ belonging to the set of real and symmetried 2 x 2 matrices and z continuous function in Đ, with continuous in D partial second derivatives with respect to x as well as functions u and v defined assuming $u(t,x) = t ? (x_{1}² + x_{2}² - 1)$ and $v(t,x) = 0$ for every $(t,x) ∈ D$.