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Asymptotic behavior of solutions to viscous conservation laws with slowly varying external forces. (English) Zbl 1184.35263

The author considers the existence and large-time behavior of solutions to the convection-diffusion equation \[ u_t-\Delta u+ b()\cdot\nabla(u|u|^{q-1})= f(x,t), \] where the forcing term \(f(x, t)\) is slowly decaying in \(t\) and \(q\geq 1+{1\over n}\) (or in some particular cases \(q\geq 1\)). The initial condition \(u_0\) belongs to in an appropriate \(L^p\) space. The author proves uniform and nonuniform decay of the solutions depending on the data and on the forcing term. The proof is based on energy estimates obtained by the Fourier-splitting technique. The novelty lies in the fact that the most of previous authors considered only zero or time-independent forcing term \(f\).

MSC:

35Q35 PDEs in connection with fluid mechanics
76R99 Diffusion and convection
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