Large time decay of solutions to Burgers type equations

Abstract

We study the initial value problem for the nonlinear dissipative equations

where a ∈ R ⁿ , ρ > 1, σ > 0. We prove that if

and

then in the case σ = (ρ − 1)/(n + 1) solutions exist globally and decay as

And in the case 0 < σ < (ρ − 1)/(n + 1), σ is close to (ρ − 1)/(n + 1), solutions have estimates

Keywords:

• Dissipative nonlinear evolution equation
• Large time asymptotics
• Burgers type equation
• 1991 Mathematics Subject Classifications: Primary 35Q35