The Behaviour of Solutions to Degenerate Double Nonlinear Parabolic Equations

We consider local behaviour of solutions to degenerate double nonlinear parabolic equations, where weight function is replaced with a double condition which supports a Poincare inequality. We give Harnack’s inequality for certain degenerate of double nonlinear parabolic equations. We used is well known that Moser’s tecnique is essentially based on the combination of a Sobolev and a Caccioppoli type inequalities. We also is established the local Holder continuity of a weak solution is a consequence of the Harnack’s inequality. However, due to the nonlinearity of the term \(\frac{\partial \left( u^{p-1} \right) }{\partial t}\) when \(p\ne 2\), it is not clear for the double nonlinear equations.