The Weak Solutions to Doubly Nonlinear Diffusion Equation with Convection Term

where 3 > q > 1. According to the different exponents of m, p, we have the following classical terminologies about the equation (1). (i) The case p = 2,m = 1, is the ordinary semilinear diffusion equation. (ii) The case p = 2,m ̸= 1, is the porous media equation, it is degenerate at u = 0 for m > 1 and singular at u = 0 for 0 < m < 1. (iii) The case p ̸= 2,m = 1, is the p-diffusion equation, it is degenerate at ∇u = 0 for 2 < p < ∞ and singular at ∇u = 0 for 1 < p < 2. (iv) The case p ̸= 2,m ̸= 1, is the doubly nonlinear diffusion equation, singularity and degeneracy at u = 0 and ∇u = 0, respectively, occur in arbitrary combinations. (v) If m(p − 1) > 1(= 1, < 1), then equation (1) is called the slow (normal, fast) diffusion equation respectively. Equation (1) appears in a number of different physical situations [1]. For example, in the study of water infiltration through porous media, Darcy’s linear relation