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On a class of scalar conservation laws with locally unbounded solutions. (English) Zbl 0717.35054

Differential equations and their applications, Proc. 7th Conf., Equadiff 7, Prague/Czech. 1989, Teubner-Texte Math. 118, 204-207 (1990).
[For the entire collection see Zbl 0704.00019.]
This note is concerned with the existence and uniqueness of solutions to the Cauchy problem for scalar conservation laws of the kind \[ (1)\quad \partial_ tu+\partial_ x[\phi (x,u)]=0\text{ in } {\mathbb{R}}_+\times {\mathbb{R}},\quad u(0,x)=u_ 0(x)\text{ in } {\mathbb{R}}, \] where \(\phi (0,u)=0\). As a matter of fact, it is easy to see that the Cauchy problem for the equation \[ (2)\quad \partial_ tu-\partial_ x(xu^ 2)=0\text{ in } {\mathbb{R}}_+\times {\mathbb{R}} \] has solutions which blow up in finite time at \(x=0.\)
As (2) suggests, the problem (1) can be investigated in weighted spaces. This approach enables to prove the existence and uniqueness of weak entropy solutions to (1), if the nonlinearity \(\phi\) is of the form \[ (3)\quad \phi (x,u)=-\sum^{n}_{k=1}| x|^{m_ k- 1}x(u^{p_ k}/p_ k). \]

MSC:

35L65 Hyperbolic conservation laws
35D05 Existence of generalized solutions of PDE (MSC2000)

Citations:

Zbl 0704.00019