Doubly nonlinear equation involving p(x)-homogeneous operators: local existence, uniqueness and global behaviour

2019/12/05 by Arora, Rakesh, Giacomoni, Jacques, Warnault, Guillaume
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.02510

Abstract

In this work, we investigate the qualitative properties as uniqueness, regularity and stabilization of the weak solution to the nonlinear parabolic problem involving general p(x)-homogeneous operators: \ \beginalignedat2 (q)/(2q-1)∂_t(u^2q-1) -∇. a(x, ∇ u) amp; = f(x,u) + h(t,x) u^q-1 amp;amp; in (0,T) × Ω; u amp; gt; 0 amp;amp; in (0,T) × Ω; u amp; = 0 amp;amp; on (0,T) × ∂Ω; u(0,.)amp;= u₀ amp;amp; in Ω. \endalignedat . Thanks to the Picone's identity obtained in [10], we prove new results about comparison principles which yield a priori estimates, positivity and uniqueness of weak solutions.

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