Martin Lazar
University of Zagreb
martin@math.hu

H-measures applied to parabolic equations

Abstract: Since their introduction, $H$-measures have been mostly used in problems related to hyperbolic equations and systems. In this study we give an attempt to apply the $H$-measure theory to parabolic equations. Through a number of examples we try to present how the differences between parabolicity and hyperbolicity reflect in the theory.

to clarify the problem, we start with the simplest example, the heat equation. It is sown that, unlike to hyperbolic equations, there is no propagation of energy ($H$-measure) along the characteristics. In order to avoid the trivial problem in which $H$-measure turns out to be zero, we introduce a new term on the right hand side of the equation. To be more precisely, we study the sequence of problems:

$$u'_n 0 \Delta u_n = -{\rm div}f_n$$

$$u_n(0) - u^0)=_n,$$

where $f_n \rightharpoonup 0$ in $L^2_{\rm loc}(R^{d+1})$. The goal is to obtain the relation between the $H$-measure associated to the sequence $f_n$, and the $H$-measure associated to $\nabla u_n$, where $u_n$ may be the unknown solution.