Heat Equation on $[0,T] \times \mathbb{R}^n$

Question

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain.

Do you know where I can find a complete proof of such a result : There exists a unique solution $u$ to the problem

$\left\{ \begin{array}{rcll} \partial_t u(t,x) - \eta \Delta u(t,x) &=& f(t,x)&\qquad\text{on } ]0,T]\times \mathbb{R}^n \\ u(0,x) &=& u_0(x)&\qquad\text{on } \mathbb{R}^n \end{array} \right.$

with $f\in L^2(0,T;L^2(\mathbb{R}^n))$, $u_0 \in H^1(\mathbb{R}^n)$ and $\eta>0$ fixed and furthermore $u\in L^2(0,T;H^2(\mathbb{R}^n))$ and $\partial_t u \in L^2(0,T;L^2(\mathbb{R}^n))$.

Thank you

Answer 1

Fourier transform in $x$ gets you there: $ \dot v+\eta\vert\xi\vert^2 v=g(t,\xi),\quad v(0)=v_0, $ so that $$ v(t,\xi)=e^{-t\eta \vert\xi\vert^2} v_0(\xi)+\int_0^te^{-(t-s)\eta \vert\xi\vert^2} g(s,\xi) ds. $$ Since $\iint_0^T\vert g(t,\xi)\vert^2 dtd\xi<+\infty$ and $\int\vert v_0(\xi)\vert^2 d\xi<+\infty$, you get $$ \iint_0^Te^{-2t\eta \vert\xi\vert^2} \vert \xi\vert^2\vert v_0(\xi)\vert^2 dt d\xi\le \int\frac{[e^{-2t\eta \vert \xi\vert^2}]^{t=0}_{t=T}}{2\eta}\vert v_0(\xi)\vert^2 d\xi\le (2\eta)^{-1}\int\vert v_0(\xi)\vert^2 d\xi<+\infty, $$ and similarly with $H=\mathbf 1_{\mathbb R_+}$ $$ \vert \xi\vert^2\int H(t-s)e^{-(t-s)\eta \vert \xi\vert^2} g(s)H(s) ds= \vert \xi\vert^2 H(t)e^{-t\eta \vert \xi\vert^2} \ast g(t)H(t) $$ with $L^2$ norm (in $t$) bounded above by $$ \vert \xi\vert^2 (\int_0^T\vert g(t,\xi)\vert^2 dt)^{1/2}\int_0^{+\infty}e^{-t\eta \vert \xi\vert^2} dt =\eta^{-1}(\int_0^T\vert g(t,\xi)\vert^2 dt)^{1/2} $$ and the square of the $L^2$ norm (in $t,\xi$) is bounded above by $$ \eta^{-2}\iint_0^T\vert g(t,\xi)\vert^2 dtd\xi<+\infty. $$ The last statement on $\dot v$ comes from the equation and the previous results.

Answer 2

See Linear and quasilinear equations of parabolic type, 1968, by O.A. Ladyzenskaja, V.A. Solonnikov, and N.N. Ural'ceva, Theorem 6.1 in Chapter III.

See also Theorem 9.1, Chapter IV, in the same book, for integrability exponents different than 2:

For $q >1$, if $f\in L^q(0,T\times \Omega)$ and $u_0\in W^{2-2/q,q}$, there exists a solution $u$, moreover $u\in L^q(0,T;W^{2,q}(\Omega))$ and $\partial_tu\in L^q(0,T;L^q(\Omega))$.