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In the book on PDEs by L. Evans, a solution to the heat equation with Dirichlet boundary conditions: $$\tag{HP} \begin{cases}\displaystyle \frac{\partial u}{\partial t}=\Delta u & x\in U,\ t\in(0, T)\\ u=0 & \text{on } \partial U\\ u=g\in L^2(U) & \text{at time }t=0 \end{cases} $$ is constructed by means of the Galerkin method. This happens to be a weak solution in the sense that $$ u\in L^2(0, T; H^1_0(U)),\quad u'\in L^2(0, T; H^{-1}(U)) $$ (the time derivative is taken in distributional sense $^{[1]}$) and it satisfies the equation where the Laplacian is taken in the "weak sense" of elliptic theory $^{[2]}$: $$\tag{1} \int_U \frac{\partial u}{\partial t}(x, t)v(x)\, dx = -\int_U \nabla u (x, t)\cdot\nabla v(x)\, dx,\quad \forall v \in H^1_0(U). $$ (see §7.1 of the second edition).

Later in the same book, (HP) is treated by means of the semigroup approach. Namely, it is proved that the unbounded operator $A=\Delta$ defined on the domain $D(A)=H^2(U)\cap H^1_0(U)$ generates a contraction semigroup $S_t$ on $L^2(U)$ space. Therefore the function $$\tag{2}u(x, t)=(S_tg)(x)$$ is a strong solution to (HP) if $g\in D(A)$. (see §7.4).

Question. Formula (2) makes sense even if $g\notin D(A)$. Is it true that in this case the function $u$ defined by (2) is a weak solution in the sense of formula (1)?

In the book Vector-valued Laplace transforms and Cauchy problems by V.A., 2nd edition $^{[3]}$, this kind of solutions are called mild solutions (Definition 3.1.1 - see also Proposition 3.1.9). This explains the title.


Notes.

$^{[1]}$ Explicitly, $$-\int_0^T u(t)\phi'(t)\, dt= \int_0^t u'(t)\phi(t)\, dt,\quad \forall \phi\in C^{\infty}_c(0, T).$$ This is somewhat related to this other question on vector-valued distributions.

$^{[2]}$ Also called energetic extension of the Laplacian.

$^{[3]}$ Recommended to me some time ago by the user lvb in this great answer.

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The easiest way to show the coincidence of the two solutions is to verify that both are weak solutions of the class $L^2(0, T; H^1_0(U))$. For problem (HP), a function $u\colon\, (0,T)\to H^1_0(U)$ is called a weak solution of the class $u\in L^2(0, T; H^1_0(U))$ if $u$ satisfies the integral identity \begin{align*} -\!\!\int\limits_{Q_T}\! uv_t\,dxdt+\int\limits_{Q_T} \nabla u\cdot\nabla v\,dxdt= \!\int\limits_U g(x)v(x,0)\,dx\tag{$\ast$}\\ \forall\, v\in H^1(Q_T)\colon\,v|_{\partial U}=0,\;v(x,T)=0, \end{align*} where $Q_T=U\times (0,T)$. For the linear problem (HP), uniqueness of the weak solution $(\ast)$ is rather obvious. Indeed, take $g=0$ and notice that the choice of a test function $$ v(x,t)=\int\limits_t^T u(x,s)\,ds $$ implies that $$ \int\limits_{Q_T}\! |u(x,t)|^2\,dxdt+\frac{1}{2}\int\limits_{U} \Bigl|\int\limits_0^T\nabla u(x,s)\,ds\Bigr|^2dx=0, $$ i.e., $u=0$ a.e. in $Q_T\,$.

To verify that a weak solution $(1)$ will be a weak solution $(\ast)$, notice that the weak solution $u$ in the sense $(1)$ satisfies the integral identity $$ \int\limits_{Q_T} u_t v \, dxdt +\int\limits_{Q_T} \nabla u \cdot\nabla v\, dxdt=0 \tag{$\ast\ast$} $$ for all test functions $v=v(x,t)$ stepwise w.r.t. variable $t$, hence for all $v\in L^2(0, T; H^1_0(U))$, and hence for all $$ v\in H^1(Q_T)\colon\,v|_{\partial U}=0,\;v(x,T)=0. $$ Integrating by parts in $(\ast\ast)$ results in the identity $(\ast)$.

To verify that the mild solution will be a weak solution $(\ast)$ multiply the equation $$ \Delta\!\!\int\limits_0^t u(x,s)\,ds=u(x,t)-g(x) $$ by $v_t(x,t)$ with $v\in H^1(Q_T)\colon\,v|_{\partial U}=0,\;v(x,T)=0$, and then integrate over $Q_T$ to find $$ \int\limits_{Q_T}\Bigl(\Delta\!\int\limits_0^t u(x,s)\,ds\Bigr)v_t(x,t)\,dxdt= \int\limits_{Q_T}uv_tdxdt+\int\limits_{U}g(x)v(x,0)\,dx, $$ where the left hand side equals \begin{multline*} \int\limits_{U}dx\int\limits_0^T v_t(x,t)\,dt\int\limits_0^t\Delta u(x,s)\,ds= \int\limits_{U}dx\int\limits_0^T\Delta u(x,s)\,ds\int\limits_s^T v_t(x,t)\,dt\\ =-\int\limits_{Q_T}\Delta u(x,s)v(x,s)\,dxds= \int\limits_{Q_T}\nabla u(x,s)\cdot\nabla v(x,s)\,dxds \end{multline*} which results in the identity $(\ast)$.

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  • $\begingroup$ +1: I find this answer to be very instructive. Is the terminology "very weak solution" used for the solution in form (*)? I recall having heard this terminology in some talk. $\endgroup$ – Giuseppe Negro Mar 19 '14 at 9:48
  • $\begingroup$ @Giuseppe Negro: For a heat-like equation, within the $L_2$-theory of weak locally integrable solutions to initial boundary value problems, there are only three types of weak solutions. Namely, 1)$\,u_t\,,\!\nabla_x u\in L^2(Q_T)$; 2)$\,u,\!\nabla_x u\in L^2(Q_T)$; 3) $\,u\in L^2(Q_T)$, with the latter type tagged as a "very weak solution". Of course, there are some additional options related to the fractional order of smoothness, to say nothing of the locally non-integrable weak solutions. $\endgroup$ – mkl314 Mar 19 '14 at 20:50

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