Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

Abstract

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over ℝd{\mathbb{R}^{d}} and in Lp{L^{p}}-spaces with respect to tight evolution systems of measures. Here, the linear part of the equation is a nonautonomous second-order elliptic operator with unbounded coefficients defined in I×ℝd{I\times\mathbb{R}^{d}}, (I being a right-halfline). To the above Cauchy problem we associate a nonlinear evolution operator, which we study in detail, proving some summability improving properties. We also study the stability of the null solution to the Cauchy problem.

Keywords

• nonautonomous second-order elliptic operators
• semilinear parabolic equations
• unbounded coefficients
• hypercontractivity
• supercontractivity
• ultraboundedness
• stability
• 35k58
• 37l15