Consider the conservation law

${\displaystyle u_{t}+f(u)_{x}=0,\quad {\text{ in }}\mathbb {R} \times (0,\infty ),}$ where ${\displaystyle f\in C^{1}(\mathbb {R} )}$.

(a) Define an integral solution of the PDE

(b) Derive the jump (Rankine-Hugoniot) condition satisfied by a piecewise smooth integral solution ${\displaystyle u}$ across a ${\displaystyle C^{1}}$ curve where this ${\displaystyle u}$ has a discontinuity.

(c) Find an integral solution to the PDE when ${\displaystyle f(u)=u^{2}+u}$ with ${\displaystyle u(x,0)=1}$ if ${\displaystyle x<0,}$ ${\displaystyle u(x,0)=-3}$ if ${\displaystyle x>0}$.