Partial Differential Equations and Characteristics — Examples

The characteristic ODE is $X^{\mathrm{\prime }}(s)=b$, so the characteristic through $x$ at time $t$ is $X(s;t,x)=x+(s-t)b$, and in particular $X(0;t,x)=x-bt$.

Apply [L1] with the characteristic from step 1.1; the transport solution is therefore $u(x,t)=u_0(X(0;t,x))=u_0(x-bt)$.

The characteristics are the straight lines $X(s;t,x)=x+(s-t)b$, so $X(0;t,x)=x-bt$, and along such a curve the source becomes $q(X(s;t,x)-bs)=q(x-bt)$, which is independent of $s$.

Applying [L1] with $c\equiv 1$ gives $u(x,t)=e^{-t}{u}_0(x-bt)+{\int }_0^t{e}^{-(t-s)}q(x-bt)ds$, and the integral equals $(1-e^{-t})q(x-bt)$, so the displayed closed form follows.

Regard the equation as transport with characteristic ODE $X^{\mathrm{\prime }}(s)=X(s)$; its solutions are the rays $X(s)=e^s{x}_0$, and by [L1] the restricted function $v(s)=u(X(s))$ satisfies $v^{\mathrm{\prime }}(s)=v(s)$, hence $v(s)=e^{s}v(0)$.

Writing $x=r\omega$ with $r=e^s$ and $\omega \in S^{n-1}$ gives $u(r\omega )=ru(\omega )$, so every solution has the claimed form with $g(\omega )=u(\omega )$; conversely, for $u(r\omega )=rg(\omega )$ the radial derivative is $u\mathrm{/}r$, hence $x\cdot \mathrm{\nabla }u=u$.

For $u_t+u_x=0$, the space-time transport vector is $(1,1)$, which is tangent to $\mathrm{\Sigma }$, so $\mathrm{\Sigma }$ is characteristic rather than noncharacteristic by [L2], and [L1] does not apply.

Every $C^1$ function of the form $u(x,t)=F(x-t)$ solves $u_t+u_x=0$, and on $\mathrm{\Sigma }$ one has $x-t=0$, so the restriction is the constant $F(0)$. Taking $F_0(s)=0$ and $F_1(s)=s$ gives two different local classical solutions with the same constant data on $\mathrm{\Sigma }$, proving nonuniqueness. On the other hand, every classical solution restricts to a constant on $\mathrm{\Sigma }$, so nonconstant prescribed data such as $g(t)=t$ are incompatible.

For $u_{xx}+u_{yy}$ one has $A=C=1$ and $B=0$, so [L2] gives $\mathrm{\Delta }=-1<0$ and the Laplace operator is elliptic; for $u_{tt}-u_{xx}$, the principal polynomial in $(\tau ,\xi )$ is ${\tau }^2-{\xi }^2$, which has two distinct real roots $\tau =\pm \xi$ for $\xi \mathrm{\ne }0$, so [L1] makes it hyperbolic.

For the heat operator ${\mathrm{\partial }}_t-{\mathrm{\Delta }}_x$, the spatial quadratic form is $\mathrm{\mid }\xi {\mathrm{\mid }}^2$, which is positive definite, while the time derivative is first order, so [L1] identifies it as parabolic.

The principal quadratic form is ${\xi }_x^2-{\xi }_y^2$, so the characteristic covectors are proportional to $d(x+y)$ and $d(x-y)$, and [L2] shows that using $\xi =x+y$ and $\eta =x-y$ follows the two characteristic families.

In these coordinates, ${\mathrm{\partial }}_x={\mathrm{\partial }}_{\xi }+{\mathrm{\partial }}_{\eta }$ and ${\mathrm{\partial }}_y={\mathrm{\partial }}_{\xi }-{\mathrm{\partial }}_{\eta }$, so $u_{xx}-u_{yy}=({\mathrm{\partial }}_{\xi }+{\mathrm{\partial }}_{\eta }{)}^{2}u-({\mathrm{\partial }}_{\xi }-{\mathrm{\partial }}_{\eta }{)}^{2}u=4u_{\xi \eta }$; this is the mixed canonical form, equivalent to the hyperbolic normal form from [L1] after a further linear recombination of $\xi$ and $\eta$.

Here $A=y$, $B=0$, and $C=1$, so [L1] gives $\mathrm{\Delta }(y)=0-y\cdot 1=-y$.

If $y>0$, then $\mathrm{\Delta }<0$ and the equation is elliptic; if $y<0$, then $\mathrm{\Delta }>0$ and it is hyperbolic; and if $y=0$, then $\mathrm{\Delta }=0$ and the principal part has rank one, so this pointwise change of sign is exactly the mixed-type behavior singled out in [L2].

By [L1], the single operator $y{u}_{xx}+u_{yy}$ is elliptic on one open region, hyperbolic on another, and degenerate on the separating line, so even within second order one operator need not belong globally to just one of the three names.

The biharmonic equation ${\mathrm{\Delta }}^{2}u=0$ has order $4$, so [L2] says the second-order trichotomy does not classify it at all; the claimed global taxonomy therefore fails both because some operators change type from point to point and because others lie outside the stated second-order scope.