Weak solutions of a conservation law with memory in several space variables (Q1378422)

The author establishes the existence and uniqueness of the weak solution of a Cauchy problem associated to a conservation law in several space variables, whose non-homogeneous term is a ``memory'' term. The existence is obtained, in a particular case, by proving the uniform boundedness of the approximate solutions in the BV-norm, and then applying the relative compactness framework, in the general case, in the \(L^1\)-framework. The uniqueness is obtained by means of a suitable modification of Kruzkov's method.