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@WillieWong I was wondering, since the matrices $A$ are diagonal, isn't there a way to apply the method of the characteristics? The equations are not uncouples since $b$ depends on all the solutions of course
Thanks @WillieWong, well asking for more regularity on $b$ is not an issue, what worries me is the local character of the result. Still do you have the complete name of Kato's article/book? I don't find it
