Abstract.
We obtain existence results for some strongly nonlinear Cauchy problems posed in \( {\mathbb{R}^{N} } \) and having merely locally integrable data. The equations we deal with have as principal part a bounded, coercive and pseudomonotone operator of Leray-Lions type acting on \( L^{P} {\left( {0,\,T;\,W^{{1,p}}_{{{\text{loc}}}} {\left( {\mathbb{R}^{N} } \right)}} \right)} \), they contain absorbing zero order terms and possibly include first order terms with natural growth. For any p > 1 and under optimal growth conditions on the zero order terms, we derive suitable local a-priori estimates and consequent global existence results.
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Leoni, F., Pellacci, B. Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data. J. evol. equ. 6, 113–144 (2006). https://doi.org/10.1007/s00028-005-0234-7
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DOI: https://doi.org/10.1007/s00028-005-0234-7