Large time decay estimates of the solution to the Cauchy problem of doubly degenerate parabolic equations with damping

In this paper we study the large time behaviour of the solution and compactification of support to the Cauchy problem for doubly degenerate parabolic equations with strong gradient damping. Under the suitable assumptions on the structure of the equation and data of the problem we establish new sharp bound of solutions for a large time. Moreover, when the support of initial datum is compact we prove that the support of the solution contains in the ball with radius which is independent in time variable. In the critical case of the behaviour of the damping term the support of the solution depends on time variable logarithmically for a sufficiently large time. The main tool of the proof is based on nontrivial use of cylindrical Gagliardo–Nirenberg type embeddings and recursive inequalities. The sup-norm estimates of the solution is carried out by modified version of the classical method of De-Giorgi–Ladyzhenskaya–Uraltseva–DiBenedetto. The approach of the paper is flexible enough and can be used when studying the Cauchy–Dirichlet or Cauchy–Neumann problems in domains with non compact boundaries.