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In this paper we study the following scalar conservation law:
It is easy to see that weak solution of (1.1) is the weak solution of the following problem:
and at
where
Kruzkov proved that if F is continuous in u and
When
Now the question is " Whether the solution obtained from Diehl can be represented by a contraction semigroup in
By looking at Diehl\'s work it is not clear that solution can be represented by a contraction semigroup.The main difficulty is to obtain a proper entropy condition at x = 0.
In this paper under the following condition:
(H) f, g are strictly convex and super linear growth, we settle this question affirmitively for arbitrary bounded initial data. Also we show that ,in general,our solution differ from the solution obtained by Diehl. Here we give an explicit formula for the solution of (1.2) satisfying (1.3). This agrees with the Lax-Olenik formula when f = g. Also we give a correct entropy condition at x = 0 so that the problem (1.2) and (1.3) admits a unique solution determined by the initial condition like in Kruzkov.