Cauchy problem for nonlinear parabolic equations with a gradient term

We consider the Cauchy problem of the nonlinear diffusive Hamilton–Jacobi equationut−Δum=|Duq|σ,ut−Δum=|Duq|σ, where m>0m>0. The existence and nonexistence of local and global solutions are studied by a priori estimates and compactness methods. We prove that the Cauchy problem of this equation has no nontrivial nonnegative global solutions if 0<σ<σ⁎0<σ<σ⁎ (here σ⁎=Nm+2qN+1); and there exist nontrivial global solutions for small initial values if σ>σ⁎σ>σ⁎. Moreover, we obtain the global existence of solutions in the limiting case and the exact L∞L∞-estimates for local and global solutions.