Understanding what people mean in PDEs

In general, the sense of solution (and the space) is a very delicate topic. Maybe, this is the reason for some authors to avoid a detailed explanation in a textbook.

Let me point out the difficulty of finding a solution of a linear pde. If a ODE has a finite dimensional space of solutions, the vector space of solutions oa given pde has infinitely many dimensions. Of course, with a nonlinear pde the compactness problem is even harder.

I will try to explain it with an example. There are "many" different concepts of solutions. Let's consider the equation $$ y''(t)=-y, y(-\pi)=y(\pi)=0. $$ The solution $y(t)=\sin(x)$ is a classical solution because it verifies the boundary conditions (in the classical sense, i.e., by evaluating) and verifies the equation in the classical sense (i.e. by taking the classical derivatives and ealuation at every point).

Let's denote the Fourier transform by $\hat{v}$. Then, we can define the space $$ H=\{v | \int_{-\pi}^\pi (1+|\xi|^2)\hat{v}^2dx\}. $$ We can multiply the equation by $v$ and integrate by parts. We get $$ \int u'v'=\int uvdx\;\;\forall v\in H. $$ Well, if $u$ verifies the previous integral equality for every $v$ in $H$, we call that function a weak solution. It's a weak solution because in general it has not the required regularity to verify the equation pointwise in the classical sense. Of course, without a minimum regularity the boundary conditions doesn't make sense too.

There are some more sense of solution. These could be related with uniqueness. For instance, if we consider the equation $$ |y'|=1,y(0)=y(1)=0, $$ we see that there is not a unique (weak) solution (think on different triangles). It's now when we use what is called "viscosity" solution. But I think that this is a quite long response, so I will stop here.

Even if the previous ideas are stated for boundary values of an ode (thus, some sort of natural analogous of elliptic pse), these ideas remains the same for evolutionary pde. For an evolutionary pde, the usual approach, is to regularize (in order one can find approximate solutions by ode techniques in Banach spaces), and then one finds estimates that ensures a minimum time of existence and a maximum growth of the (spatial) norm (or in general "energy"). With these ingredients (the approximate solutions + the good estimates) one invokes some compactness theorem and verify that the limit verifies the equation according to the classical/weak/distributional sense.

Hopefully this helps you a little bit.