On the asymptotic behaviour of solutions of some nonlinear elliptic and parabolic equations
The paper deals with the nonlinear differential equations of the form \[ \Delta u + \frac{1}{q-1} \nabla (c_i | u| ^{q-2} u) = 0, \qquad (1 <q < 4) \tag{1} \] and \[ \Delta u + \frac{1}{q-1} \nabla (c_i | u| ^{q-2} u) = k\dot u, \qquad (1 < q \leq 2), \tag{2} \] which are related with the processes of diffusion-convection. Here \(u = u(x)\) with \(x = (x_1,x_2,x_3) \in D \times [0, \infty)\) in (1) and \(u = u(x,t)\) with \(x = (x_1,x_2,x_3) \in D \times [0, \infty),\) \linebreak \(t \in (0,T)\) in (2). The plane domain \(D\) is supposed to be sufficiently smooth and \(0 < T \leq \infty\) is given. Assuming boundary conditions of Dirichlet type in (1) together with an initial condition in (2) and introducing the function \(\Phi(z)\) for (1) or \(\Phi(z,t)\) for (2) defined by means of the integral of the solution over the domain \(\{(x_1,x_2) \in D, x_3=z\}\) the author proves several estimates for these functions. As the author has remarked, some results given in [\textit{C. O. Horgan, L. E. Payne}, Arch. Ration. Mech. Anal. 122, No. 2, 122--144 (1993; Zbl 0790.31002)] are used in the paper.
scientific article; zbMATH DE number 1880008
scientific article