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$$L^ 1$$-contraction and uniqueness for quasilinear elliptic-parabolic equations. (English) Zbl 0862.35078
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.

##### MSC:
 35M10 PDEs of mixed type 35G30 Boundary value problems for nonlinear higher-order PDEs
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