Given the diffusion equation:
$$\dfrac{\partial\Phi(x,t)}{\partial t}=k\dfrac{\partial^2\Phi(x,t)}{\partial x^2}$$ and the condition: $$\int_{-\infty}^{+\infty}|\Phi(x,t)|^2dx=1,\forall t\in(-\infty,+\infty)$$ what is the method to find the solutions of the previous partial differential equation? Thanks
Suppose such a solution exists. Then due to the uniqueness of the Cauchy problem $$ \left\{ \begin{array}{rcl} u_t-ku_{xx}&=&0,\ t>0,\\ u|_{t=0}&=&\Phi(x,0), \end{array} \right. $$ solution in $L_\infty([0,T],L_2(\mathbb R))$, $0<T<\infty$, the solution can be represented as a Poisson potential: $$ \Phi(x,t)=\int_{\mathbb R}Z(x-y,t)\Phi(y,0)\,dy. $$ Нere $Z(x,t)=(4\pi kt)^{-1/2}e^{-x^2/4kt}$ is the fundamental solution of the heat equation.
Form there it's straightforward to check that for $t>0$ derivatives $\Phi_t(\cdot,t),\ \Phi_{xx}(\cdot,t)\in L_2(\mathbb R)$. Denote $$ I(t)=\int_{-\infty}^{+\infty}\Phi^2(x,t)\,dx. $$ Then $$ I'(t)=2\int_{-\infty}^{+\infty}\Phi_t\Phi\, dx=2\int_{-\infty}^{+\infty}k\Phi_{xx}\Phi\, dx=-2k\int_{-\infty}^{+\infty}\Phi_x^2\, dx<0. $$ Therefore $I(t)$ cannot be a constant.