Existence and Uniqueness of Energy Weak Solutions to Boundary and Obstacle Problems for Quasilinear Elliptic-Parabolic Equations

. We prove existence and uniqueness of weak solutions to initial-boundary value problem and problem with inner obstacle for elliptic-parabolic equations @ t b(u) \Gamma divfjffi(u)j m\Gamma2 ffi(u)g = f(x; t); ffi(u) = ru+ k(b(u))~e; j~ej = 1; m ? 1 with a monotone nondecreasing function b. These equations arise in the theory of non-Newtonian filtation and mathematical glaciology. 1. Introduction Let Q T =\Omega \Theta(0; T ],\Omega ae R n ; n 1. Consider initial-boundary value problem Mu := @ t b(u) \Gamma div a(b(u); ru) = f(x; t) in Q T ; (1.1) u = 0 on S T = @\Omega \Theta(0; T ]; (1.2) b(u) = b 0 on\Omega \Thetaft = 0g (1.3) where b : R ! R be monotone nondecreasing and continuous while function a be continuous on R \Theta R n , strictly monotone in p 2 R n for any w 2 R and satisfies the growth condition ja(w; p)j m 0 c(jpj m + \Psi(w) + 1); m ? 1; 1=m + 1=m 0 = 1; (1.4) where \Psi(w) is the Legendre transform of the primitive of b, i.e., \Psi(w) = s..