Existence of solution for a class of heat equation with double criticality

Abstract

Abstract In this paper, we study the following class of quasilinear heat equations { u t − Δ Φ u = f ( x , u ) in Ω , t > 0 , u = 0 on ∂ Ω , t > 0 , u ( x , 0 ) = u 0 ( x ) in Ω , where Δ Φ u = div ( φ ( x , | ∇ φ | ) ∇ φ ) and Φ ( x , s ) = ∫ 0 | s | φ ( x , σ ) σ d σ is a generalized N-function. We suppose that Ω ⊂ R N ( N ≥ 2 ) is a smooth bounded domain that contains two open regions Ω N and Ω p with Ω ‾ N ∩ Ω ‾ p = ∅ . Under some appropriate conditions, the global existence will be done by combining the Galerkin approximations with the potential well theory. Moreover, the large-time behavior of the global weak solution is analyzed. The main feature of this paper consists that − Δ Φ u behaves like − Δ N u on Ω N and − Δ p u on Ω p , while the continuous function f : Ω × R → R behaves like e α | s | N N − 1 on Ω N and | s | p ⁎ − 2 s on Ω p as | s | → ∞ .