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A Nonlinear Heat Equation with Temperature-Dependent Parameters

A nonlinear partial differential equation of the following form is considered:
u'-\div\Big(a(u)\nabla u\Big)+ b(u)\;\vert\nabla u\vert^2=0, 
which arises from the heat conduction problems with strong temperature-dependent material parameters, such as mass density, specific heat and heat conductivity. Existence, uniqueness and asymptotic behavior of initial
boundary value problems under appropriate assumptions on the material parameters are established for one-dimensional case. Existence and asymptotic behavior for two-dimensional case are also proved.