The Support Behavior of the Solution to the Cauchy Problem for Higher Order Weighted Parabolic Equations

In this paper we study the finite speed of propagation property to the Cauchy problem for weighted higher order degenerate parabolic equations. We prove that if initial data is compactly support in some fixed ball, then so does the solution for all time. Because we are considering exponentially growing weights, the size of the support should expand more slowly over time than in the non-weighted case. We prove that for a large time the support of the solution expand with logarithmic rate. That estimate of support meets with known estimate for second order parabolic equations. The main tool of the proof is based on local energy estimates on annuli which allows us to consider even nonpower character of weights. It works even in cases when the weighted Gagliardo-Nirenberg inequality does not occur. Early that approach was utilized by D. Andreucci and by author for equations in domains with noncomact boundaries and for higher order parabolic equations including the thin film equation.