On well-posedness classes of locally bounded generalized entropy solutions of the Cauchy problem for quasilinear first-order equations

We study scalar conservation laws with power growth restriction on the flux vector. For such equations, we found correctness classes for the Cauchy problem among locally bounded generalized entropy solutions. These classes are determined by some exponents of admissible growth with respect to space variables. We give examples showing that enlargement of the growth exponent leads to failure of the correctness.