A NOTE ON POSITIVE SOLUTIONS FOR CONSERVATION LAWS WITH SINGULAR SOURCE

作者：Amadori D^*; Coclite G M

来源：Proceedings of the American Mathematical Society, 2013, 141(5): 1613-1625.

• Singular nonlinear problems
• positive solutions
• conservation laws

We consider the Cauchy problem for the scalar conservation law %26lt;br%26gt;partial derivative(t)u + partial derivative(x)f(u) = 1/g(u), t %26gt; 0, x is an element of R, %26lt;br%26gt;with g is an element of C-1(R), g(0) = 0, g(u) %26gt; 0 for u %26gt; 0, and assume that the initial datum u(0) is nonnegative. %26lt;br%26gt;We show the existence of entropy solutions that are positive a. e. by means of an approximation of the equation that preserves positive solutions and by passing to the limit using a monotonicity argument. The difficulty lies in handling the singularity of the right-hand side (the source term) as u possibly vanishes at the initial time. The source term is shown to be locally integrable. %26lt;br%26gt;Moreover, we prove a uniqueness and stability result for the above equation.

• 出版日期2013-5