Extremal solutions for quasilinear parabolic systems in trapping regions

Siegfried Carl (1) , Dumitru Motreanu (2)

1 MLU - Martin-Luther-Universität Halle Wittenberg
2 LAMPS - LAboratoire de Modélisation Pluridisciplinaire et Simulations

We consider the initial-Dirichlet boundary value problem for quasilinear parabolic systems in a cylindrical domain $Q = n \times (0,\tau)$ of the form ($i = 1,2$) $$\frac{\partial u_i}{\partial t}-\Delta_{p_i}u_i=f_i(x,t,u_1,u_2,\nabla u_1,\nabla u_2)\quad\text{in}\quad Q$$ with a diagonal $(p_1, p_2)$-Laplacian as leading elliptic operator, and with a lower order vector field $f=(f_1,f_2)$ that may depend also on the gradient of the solution $u=(u_1,u_2)$. We establish an enclosure and existence result for weak solutions in terms of trapping regions which stand for rectangles formed by pairs of appropriately defined sub-supersolutions, and prove the existence of extremal solutions within trapping regions without imposing any monotonicity conditions on the lower order vector field. Finally, we provide conditions that allow us to construct trapping regions. It should be noted that the results obtained in this paper may be extended to more general quasilinear systems, where the $p_i$-Laplacian is replaced by a general divergence form Leray-Lions operator div $A_i(x,t,u_i,\nabla u_i)$.

Domaines: Mathématiques [math]

Siegfried Carl, Dumitru Motreanu. Extremal solutions for quasilinear parabolic systems in trapping regions. Pure and Applied Functional Analysis, 2018, 3 (1), pp.57-74. ⟨hal-01900576⟩