Elliptic and Parabolic Equations - Abstract

Namlyeyeva, Yuliya

Isolated singularities of solutions of nonlinear anisotropic elliptic and parabolic equations

We consider wide classes of quasilinear anisotropic elliptic equations and doubly nonlinear anisotropic parabolic equations, whose solutions have singularity at a point. The model representatives of these equations are the following $$sumlimits_i=1^n left( u_x_i ^p_i-2 u_x_iright)_x_i=0,$$$$ u_t - sum^n_i=1 left( u ^(m_i-1)(p_i-1) left u_x_iright ^p_i-2 u_x_iright)_x_i = 0,$$ where $1 < p_1leqslant p_2leqslantdotsleqslant p_n.$ We establish sharp pointwise conditions for removable isolated singularity of solutions of such equations. The precise upper and lower estimates near the singularity point for solutions with non-removable point singularity are obtained.
This talk represents joint work with I.I. Skrypnik and A.E. Shishkov.