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EXISTENCE RESULTS FOR L1 DATA OF SOME QUASI-LINEAR PARABOLIC PROBLEMS WITH A QUADRATIC GRADIENT TERM AND SOURCE

F. ANDREU, S. SEGURA DE LÉON, L. BOCCARDO, L. ORSINA

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In this paper we deal with a Cauchy–Dirichlet quasilinear parabolic problem containing a gradient lower order term; namely, ut - Δu + |u|2 γ-2u |∇u|2 = |u|p-2u. We prove that if p ≥ 1, γ ≥ ½ and p < 2 γ + 2, then there exists a global weak solution for all initial data in L1 (Ω). We also see that there exists a non-negative solution if the initial datum is non-negative.