# Well-posedness for Heat Equation with Robin Boundary Condition

Can anyone help me prove the well-posedness of the following heat equation with Robin boundary condition?

$$u_t(x,t)=u_{xx}(x,t)$$

$$u(0,t)=0$$

$$u_x(1,t)=-au(1,t)$$

where $$a>0$$.

The existence of the solution may be simply obtained by separation of variable. Are there any good references on this problem?

your problem can be written under the form $$u'(t)=Au\\u(0)=u_0$$ where $$A:D(A) \subset L^2(0,1) \to L^2(0,1)$$ $$D(A)=\left\lbrace v\in H^1(0,1), v(0)=0, v_x(1)+av(1)=0\right\rbrace$$ It is straitforward to see that $$(Au,u)_{L^2(0,1)} \ \leqslant 0$$, and $$I-A$$ is maximal, ($$A$$ is the second derevative), therefore, by semigroup theory, there exists one solution in $$C(0,T;L^2(0,1)$$ (if your initial state is $$L^2(0,1)$$)
Your problem is missing an initial condition such as $$u(x,0)=u_0(x)$$, which is necessary. Assuming $$u_1,u_2$$ are two such solutions satisfying the same initial condition and conditions you have specified, then $$v=u_1-u_2$$ satisfies $$v_t = v_{xx} \\ v(x,0)=0 \\ v(0,t)=0,\;\;v_x(1,t)=-av(1,t).$$ Then, \begin{align} &\frac{d}{dt}\int_{0}^{1}v(x,t)^2dx \\ &=2\int_0^1v(x,t)v_t(x,t)dx \\ &= 2\int_0^1v(x,t)v_{xx}(x,t)dx \\ &= 2v(x,t)v_{x}(x,t)|_{x=0}^{1}-2\int_0^1 v_x(x,t)^2dx \\ &= 2v(1,t)v_x(1,t)-2v(0,t)v_x(0,t)-2\int_0^1 v_x^2x \\ &= 2v(1,t)v_x(1,t)-2\int_0^1 v_x^2dx \\ &= -2av_x(1,t)^2-2\int_0^1 v_x^2 dx \le 0. \end{align} Because $$\int_0^1v(x,t)^2dx = 0$$ for $$t=0$$, the above forces $$\int_0^1v(x,t)^2dx=0,\;\;\; t \ge 0.$$
This is enough to give $$v(x,t)=0$$ for all $$t\ge 0$$. So $$u_1=u_2$$.
• One might say that the initial condition is just arbitrary if none is chosen..I.E $u(x,0)= g(x)$. Even if it isn’t stated, it is somewhat implied, no? Commented Dec 8, 2018 at 20:26