Conditions for the Existence of Global Solutions to Doubly Nonlinear Advection-diffusion Equations

ABSTRACT

In this work, we consider a initial-value problem for an doubly nonlinear advection-diffusion equation, and we present a critical value of κ up to wich the initial-value problem has global solution independent of the initial data u ₀, and from which global solutions may still exists, but from initial data u ₀ satisfying certain conditions. For this, we suppose that the function \(<mml:math><mml:mi mathvariant="bold-italic">f</mml:mi> <mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>,</mml:mo> <mml:mi>t</mml:mi><mml:mo>,</mml:mo><mml:mi>u</mml:mi> <mml:mo>)</mml:mo></mml:math>\) in the advection term, writted in the divergent form, satisfies certain conditions about your variation in ℝⁿ , and we also use the decrease of the norm $L^1\left({\mathrm{ℝ}}^n\right)$ and an control for the norm $L^{\infty }\left({\mathrm{ℝ}}^n\right)$ of solution $u(·,t)$.

Keywords:
doubly nonlinear parabolic equation; global solutions; conditions for global solutions