Past exam of the differential equations 1 course of the University of Oxford / 2023 / 3 / b

Define the squared L2 norm $E(t)={\int }_0^{1}u(x,t{)}^{2}dx$. Multiplication of the semilinear partial differential equation by $u$, followed by integration by parts, givesAt $x=1$ the Dirichlet boundary condition makes the boundary term zero. At $x=0$ the Robin boundary condition is $u_x(0,t)=\alpha u(0,t)$, so subtraction of the lower endpoint gives $-\alpha u(0,t{)}^2$. In particular the sign is dissipative. Thus the exact energy identity iswhich implies

For every real $r$ and integer $n\ge 1$, $|\sin r|\le \min (1,|r|)$ givesThis estimate covers both even and odd $n$, without any assumption on the sign of $u$. Since $a(t)\ge 0$, the energy estimate from part (i) becomesEquivalently, an integrating factor givesThe bracket is nonnegative and starts at zero, so it vanishes identically. This is also the zero-data case of the Gronwall inequality. The vanishing L2 norm and continuity of $u$ imply