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Capacitary estimates of positive solutions of semilinear elliptic equations with absorbtion

Moshe Marcus, Laurent Véron (2004)

Journal of the European Mathematical Society

Let be a bounded domain of class in N and let be a compact subset of . Assume that and denote by the maximal solution of in which vanishes on . We obtain sharp upper and lower estimates for in terms of the Bessel capacity and prove that is -moderate. In addition we describe the precise asymptotic behavior of at points , which depends on the “density” of at , measured in terms of the capacity .

Classical solutions to the scalar conservation law with discontinuous initial data

Jędrzej Jabłoński (2013)

Colloquium Mathematicae

Sufficient and necessary conditions for the existence and uniqueness of classical solutions to the Cauchy problem for the scalar conservation law are found in the class of discontinuous initial data and non-convex flux function. Regularity of rarefaction waves starting from discontinuous initial data and their dependence on the flux function are investigated and illustrated in a few examples.

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