A new family of higher order nonlinear degenerate parabolic equations

There has been much investigation of higher order nonlinear degenerate equations of the form h(t) = (M( h) (delta H/delta h)(x))(x), whereM is a specified function and H is the quadratic first order energy functional 1/2 integral h(x)(2) dx. The energy functional arises in many physical models, but is not universal among higher order parabolic equations. Recent investigations have motivated the study of other energy functionals, such as H-p = integral( h(x)(2))(p/2) dx for p not equal 2. We undertake such a study here, proving the existence of weak solutions for appropriate boundary conditions, nonnegativity and positivity properties of solutions. Moreover, an entropy dissipation-entropy estimate for solutions of this equation is obtained. Support properties and long time behaviour of solutions are also discussed for various cases.