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On classes of well-posedness for quasilinear diffusion equations in the whole space
Discrete and Continuous Dynamical Systems-Series S ( IF 1.8 ) Pub Date : 2020-05-14 , DOI: 10.3934/dcdss.2020361
Boris Andreianov , , Mohamed Maliki , ,

Well-posedness classes for degenerate elliptic problems in $ {\mathbb R}^N $ under the form $ u = \Delta {{\varphi}}(x,u)+f(x) $, with locally (in $ u $) uniformly continuous nonlinearities, are explored. While we are particularly interested in the $ L^\infty $ setting, we also investigate about solutions in $ L^1_{loc} $ and in weighted $ L^1 $ spaces. We give some sufficient conditions in order that the uniqueness and comparison properties hold for the associated solutions; these conditions are expressed in terms of the moduli of continuity of $ u\mapsto {{\varphi}}(x,u) $. Under additional restrictions on the dependency of $ {{\varphi}} $ on $ x $, we deduce the existence results for the corresponding classes of solutions and data. Moreover, continuous dependence results follow readily from the existence claim and the comparison property. In particular, we show that for a general continuous non-decreasing nonlinearity $ {{\varphi}}: {\mathbb R}\mapsto {\mathbb R} $, the space $ L^\infty $ (endowed with the $ L^1_{loc} $ topology) is a well-posedness class for the problem $ u = \Delta {{\varphi}}(u)+f(x) $.

中文翻译:

整个空间中拟线性扩散方程的适定性类

$ {\\ mathbb R} ^ N $中退化椭圆问题的适定性类,形式为$ u = \ Delta {{\ varphi}}(x,u)+ f(x)$,其中局部(在$ u中)探索了均匀连续非线性。尽管我们对$ L ^ \ infty $设置特别感兴趣,但我们还研究了$ L ^ 1_ {loc} $和加权$ L ^ 1 $空间中的解决方案。我们给出一些充分的条件,以使关联解决方案具有唯一性和比较属性。这些条件用$ u \ mapsto {{\ varphi}}(x,u)$的连续模数表示。在$ {{\ varphi}} $对$ x $的依存关系的其他限制下,我们推导了相应类的解决方案和数据的存在结果。而且,连续的依赖结果很容易从存在声明和比较属性中得出。
更新日期:2020-05-14
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