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

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In this paper, we study the following class of quasilinear heat equations{ut−ΔΦu=f(x,u)inΩ,t>0,u=0on∂Ω,t>0,u(x,0)=u0(x)inΩ, where ΔΦu=div(φ(x,|∇φ|)∇φ) and Φ(x,s)=∫0|s|φ(x,σ)σdσ is a generalized N-function. We suppose that Ω⊂RN(N≥2) is a smooth bounded domain that contains two open regions ΩN and Ωp 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 −ΔΦu behaves like −ΔNu on ΩN and −Δpu on Ωp, while the continuous function f:Ω×R→R behaves like eα|s|NN−1 on ΩN and |s|p⁎−2s on Ωp as |s|→∞.