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i'm struggling to prove solution uniqueness with this one. thanks for the assistance.

$$u_{tt}+u_{t}=u_{xx} $$ $$u_x(0,t)=u_x(\pi,t)=0 $$ $$u(x,0)=1$$ $$u_t(x,0)=\cos^2(x)$$

Prove the uniqueness of the solution

well, i defines two solutions, $$w(x,t)=u(x,t)-v(x,t) $$ in aim to show $$w(x,t)=0 $$ and therefore $$ u(x,t)=v(x,t)$$

i multiplied each side of the equation with $w$ and integrated it, to get: $$\int_0^\pi w(w_{tt}+w_{t}) dx=\int_0^\pi ww_{xx} dx$$ $$\int_0^\pi ww_{tt}+\int_0^\pi ww_{t} dx=\int_0^\pi ww_{xx} dx$$ where, $$ww_{t}=\frac{d}{dt}\big[\frac{1}{2}w^2\big] \rightarrow \frac{d}{dt}\frac{1}{2} \int_0^\pi w^2 dx $$ by integration by part i got the follows: $$E'(t) = \frac{d}{dt}\frac{1}{2} \int_0^\pi w^2 dx = \int_0^\pi w_t^2-w_x^2 dx$$

am i in the right way? asking your as guidance

user862242
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1 Answers1

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Maybe it is better to write down an explicit energy first and do the calculations with that one. Try this definition of an energy for your $w$: $$E(w(\cdot,t)):=\int_0^\pi (w_x)^2 + (w_t)^2\, dx.$$ This $E$ is always nonnegative. Now show $\frac{d}{dt}E(w(\cdot,t))\leq 0$ and $E(w(\cdot,0))=0$ (these will need the equation, the initial and boundary data and partial integration). I leave these steps for you to complete yourself.

On another note: It seems to me, you used partial integration to infer the following: $$\int_0^\pi w w_{tt}\, dx = - \int_0^\pi (w_t)^2\, dx.$$ Since the integration is with respect to $x$ and the derivative with respect to $t$, this step is not correct.

humanStampedist
  • 3,622