## 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)

Zbl 0704.00019