engleski

H-measures applied to parabolic equations

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 shown 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: $$ \eqalign{; ; ; ; ; u_n' - \Delta u_n &= - \dv f_n \cr u_n(0) &= u_n^0 , \cr }; ; ; ; ; $$ where $f_n \dscon 0$ in $\Ldl\Rdpj$ 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.

H-measure; parabolic equations

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