I need to solve Problem 3.5 - 11 p. 164 of the book Partial Differential Equations by Lawrence C. Evans (2nd ed., AMS, 2010):
- Show that $$ u(x,t) = \begin{cases} -\dfrac{2}{3}\left(t+\sqrt{3x+t^2}\right); & \text{if } 4x + t^2 >0\\ 0; & \text{if } 4x + t^2<0 \end{cases} $$ is an (unbounded) entropy solution of $u_t + \left(\dfrac{u^2}{2}\right)_x=0$.
Clearly it is easy to see that this is both unbounded and a solution to the given PDE, however I am not sure how to gather any information to say that it satisfies the entropy condition from the solution alone. Any hints would be welcomed.
Edit. I know the definition (Evans, §3.4.3.b p. 150). A weak solution $u\in L^\infty(\Bbb R\times (0,\infty))$ of the initial value problem $u_t + \left(\dfrac{u^2}{2}\right)_x = 0$ with data $u|_{t=0} = g$ is an entropy solution if
$$ u(x+z,t) - u(x,t) \leq C \left(1 + \frac{1}{t}\right) z \tag{ii} $$ for some constant $C>0$ and a.e. $x$, $z \in \Bbb R$, $t>0$, with $z>0$.
Is it the one needed to solve the problem? How to use it?