\(L^ 1\)-contraction and uniqueness for quasilinear elliptic-parabolic equations
The following quasilinear elliptic-parabolic equation: \[ \partial_t[b(u)]- \text{div}[a(\nabla u,b(u))]+ f(b(u))=0\quad\text{in }(0,T)\times\Omega \] in \(u\) is considered, where \(b:\mathbb{R}\to\mathbb{R}\) is monotone nondecreasing and continuous and \(a:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}^n\) s.t. \(u\in H^{1,r}(\Omega)\mapsto-\text{div}[a(\nabla u,w)]\in(H^{1,r}(\Omega))\) is strictly monotone for any \(w\in\mathbb{R}\) and some \(r\in(1,\infty)\). The problem is completed by some boundary conditions for the function \(u\) and some initial values for \(b(u)\). It is known that the natural solution space for this equation is given by all \(u\) of finite energy, i.e. \[ \sup_{t\in(0,T)} \int_\Omega\Psi(b(u(t)))+ \int_{(0,T)\times\Omega}|\nabla u|^r<+\infty,\text{ where }\Psi(w):=\sup_{z\in\mathbb{R}}\Biggl(zw- \int^z_0b(\xi)d\xi\Biggr), \] is the Legendre transform of the primitive of \(b\). Under these assumptions the \(L_1\)-contraction principle and uniqueness of solutions for this quasilinear elliptic-parabolic equation are proved.
scientific article; zbMATH DE number 946923