## Global existence and decay for a nonlinear parabolic equation.(English)Zbl 0782.35009

The author studies the global existence and decay of solutions of the semilinear parabolic problem $u_ t=\Delta u+\eta|\nabla u|, \quad \eta\neq 0, \qquad u(x,0)=u_ 0(x), \quad x\in\mathbb{R}^ n.$

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations

### Keywords:

semilinear parabolic equation; global existence; decay
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### References:

 [1] Alfonsi, L.; Weissler, F., Blow up in $$R^n$$ for a parabolic equation with damping nonlinear gradient term (1991), preprint [2] Chipot, M.; Weissler, F., Some blowup results for a nonlinear parabolic equation with a gradient term, SIAM J. math. Analysis, 20, 886-907 (1989) · Zbl 0682.35010 [3] Lions, P. L., Generalized solutions of Hamilton-Jacobi equations, Pit́man Research Notes in Mathematics, Vol. 69 (1982) · Zbl 1194.35459 [4] Hörmander, L., Nonlinear hyperbolic differential equations (1988), University of Lund, Lecture Notes [5] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1977), Springer: Springer Berlin · Zbl 0691.35001
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