In this dissertation we discuss a new approach for studying forward-backward quasilinear diffusion equations. Our main idea is motivated by a reformulation of such equations as non-homogeneous partial differential inclusions and relies on a Baire's category method. In this way the existence of Lipschitz solutions to the initial-boundary value problem of those equations is guaranteed under a certain density condition. Finally we study two important cases of anisotropic diffusion in which such density condition can be realized.The first case is on the Perona-Malik type equations. In 1990, P. Perona and J. Malik \cite{PM} proposed an anisotropic diffusion model, called the Perona-Malik model, in image processing\[u_t=\dv \Big(\frac{Du}{1+|Du|^2}\Big)\]for denoising and edge enhancement of a computer vision. Since then the dichotomy of numerical stability and theoretical ill-posedness of the model has attracted many interests in the name of the Perona-Malik paradox \cite{Ki}. Our result in this case provides the model with mathematically rigorous solutions in any dimension that are even reflecting some phenomena observed in numerical simulations.The other case deals with the existence result on the H\"ollig type equations. In 1983, K. H\"ollig \cite{Ho} proved, in dimension $n=1$, the existence of infinitely many $L^2$-weak solutions to the initial-boundary value problem of a forward-backward diffusion equation with non-monotone piecewise linear heat flux, and this piecewise linearity was much relaxed later by K. Zhang \cite{Zh1}. The work \cite{Ho} was initially motivated by the Clausius-Duhem inequality in the second law of thermodynamics, where the negative of the heat flux may violate the monotonicity but should obey the Fourier inequality at least. Our result in this case generalizes \cite{Ho, Zh1} to all dimensions.
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