Partial differential equation ($F(x,t,u,u_t,u_x,u_{xx},\dots )=0$)

A partial differential equation relates an unknown function of several independent variables to its partial derivatives. For $u(x,t)$, the notation $u_t=\partial u/\partial t$ means changing time while holding position fixed; $u_{xx}={\partial }^{2}u/\partial x^2$ measures how its spatial slope changes. Unlike an ordinary differential equation, a partial differential equation can describe an entire evolving profile rather than a single trajectory.

The heat equation $u_t=D{u}_{xx}$ smooths spatial variations, the transport equation $u_t+c{u}_x=0$ moves them, and Laplace's equation $\Delta u=0$ describes a spatial equilibrium. Here $D>0$ is a diffusivity, $c$ is a velocity, and $\Delta$ is the Laplacian. The periodic heat and transport comparison makes the difference visible using identical initial conditions.

A partial differential equation usually supplies a local rule rather than a complete answer. For example, every $u(x,t)=A+Bx$ satisfies the heat equation with zero PDE source term; data select which profile is relevant. A complete problem specifies its domain of a function, coefficients, initial conditions, boundary conditions, and meaning of solution of a partial differential equation. A well-posed problem has existence, uniqueness, and continuous dependence on initial data in stated spaces; the equation alone guarantees none of these.^[ref]

For a differential equation $L[u]=0$ with its comparison order fixed, a supersolution satisfies $L[w]\ge 0$. For example, if $L={\partial }_t-{\partial }_x^2+Q$ with $Q\ge 0$, a sufficiently regular $w$ satisfying $L[w]\ge 0$ and dominating the initial and boundary data lies above a solution by the parabolic comparison principle. An unbounded-domain barrier such as $w=\eta e^{3t}(1+x^2)$ can provide the missing boundary control at large distance. The sign convention belongs to the chosen differential operator and must be stated.

A free boundary problem determines both a field and part of its domain boundary. A glacier terminus, for example, separates positive ice thickness from bare bed. Mass conservation, nonnegative thickness, a rule for ice ablation only where ice exists, and suitable interface or entropy solution conditions are needed; a formal zero of one characteristic branch alone need not give the selected physical boundary.

A drying front separates a positive liquid layer from a region where its thickness is zero. Its position is an unknown boundary determined with the wet-region solution. At a continuous zero-height front, a jump balance alone need not fix its speed; the vanishing-depth profile and removal law can supply that information. Source terms representing evaporation or infiltration must respect nonnegative water availability in the dry region.

A second-order partial differential equation is of mixed type when the type of its principal part changes across its domain. For $A{u}_{xx}+2B{u}_{xy}+C{u}_{yy}=0$, the sign of $B^2-AC$ distinguishes a hyperbolic partial differential equation, parabolic type and an elliptic partial differential equation. For example, $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ is elliptic above $y=0$, hyperbolic below it, and degenerate on the interface. On its hyperbolic side, the change $\xi =x+y$, $\eta =x-y$ reduces it to $u_{\xi \eta }=0$; above it is the Laplace equation. Type classification alone supplies no matching rule for the normal derivative.

A function continuous across an interface can have different one-sided normal derivatives. For instance, $u(x,y)=y$ when $y<0$ and $u(x,y)=0$ when $y\ge 0$ is continuous, satisfies $u_{xx}+\mathrm{sign}(y)u_{yy}=0$ on each open side, and has zero tangential second derivative on the interface. Its lower $y$-derivative is one and its upper $y$-derivative is zero. Thus the normal derivative does not exist there as a two-sided derivative.

More generally, for functions smooth on each side with one-sided first derivative traces, global $C^1$ regularity would force both traces to equal the derivative on the interface. Unequal traces therefore preclude a globally classical $C^2$ solution as well. A condition on the function value alone is not a condition on its normal derivative; a specified transmission law or an appropriate weak formulation is additional information.

A partial differential equation whose highest partial derivatives have order one. For $a(x,y)u_x+b(x,y)u_y=c(x,y,u)$, the method of characteristics follows the ordinary differential equations $\dot{x}=a$, $\dot{y}=b$, $\dot{u}=c$.

For a forward heat equation or a related reaction-diffusion equation on a spatial domain $\Omega$, the parabolic boundary is the union of the initial face and the lateral spatial boundary. It excludes the interior of the final-time face. This is the boundary relevant to the forward maximum principle: values at a later time are controlled by earlier data and the lateral boundary. Writing $\Omega \times \{0\}$ instead of $\overline{\Omega }\times \{0\}$ in the union gives the same set, since the lateral term includes the initial rim.

On $0<x<L$, $t>0$, an initial condition $u(x,0)=g(x)$ gives the starting profile. For the heat equation, Dirichlet boundary conditions $u(0,t)=a(t)$, $u(L,t)=b(t)$ can model ends held at prescribed temperatures. Alternatively, zero Neumann boundary conditions describe insulated ends in the constant-conductivity model. These choices can produce different evolutions from the same initial condition.

For a continuous classical solution reaching the corners, the Dirichlet boundary data must satisfy $g(0)=a(0)$ and $g(L)=b(0)$; higher smoothness can require further compatibility. In dimensionless variables, a complete worked heat equation problem isIts exact classical solution is $u(x,t)=x+e^{-{\pi }^{2}t}\sin (\pi x)$. The straight profile $x$ is the equilibrium; the extra bump decays. Substituting verifies the equation and every datum, and the energy estimate proves uniqueness among smooth solutions with these data.

For the transport equation with $c>0$ on the same interval, prescribe an inflow transport boundary condition only at $x=0$. Backtracking a characteristic curve gives $g(x-ct)$ if $x\ge ct$, and $a(t-x/c)$ otherwise. Values at $x=L$ are determined by this formula; independently prescribing them can make the problem inconsistent.^[ref]

On the unit square, these data determine only the zero solution of the heat equation $u_y=u_{xx}$. The squared $L^2$ norm decreases because its derivative is $-2\int u_x^{2}dx$, and its initial value is zero. For Laplace's equation, $\sin (\pi x/2)\sinh (\pi y/2)$ is a nonzero solution with the same three data; adding zero Dirichlet data on the top edge restores uniqueness by Green's first identity. For the wave equation $u_{yy}=u_{xx}$, $\sin (\pi x/2)\sin (\pi y/2)$ is a counterexample; the missing datum is initial velocity. Adding $u_y(x,0)=0$ restores uniqueness by conservation of $\frac{1}{2}\int (u_x^2+u_y^2)dx$. Thus the type of equation determines what information is needed on an unprescribed edge.

An initial-boundary value problem specifies a partial differential equation, initial conditions, and boundary conditions. For example, the half-line wave equation $u_{tt}=c^2{u}_{xx}$ requires an initial displacement and initial velocity for $x\ge 0$ together with an endpoint condition such as $u(0,t)=0$ for $t>0$. Its meaning also depends on the chosen solution class: a classical solution satisfies the equation pointwise, while a weak solution satisfies an integrated form. These data must obey the compatibility conditions needed by the chosen regularity.

A prescribed PDE source term changes the heat equation to $u_t=D{u}_{xx}+f(x,t)$, or the transport equation to $u_t+c{u}_x=f(x,t)$. If $u$ is a temperature, $f$ has units of temperature per time. The formulas below assume $f=0$; a nonzero PDE source term can change the spatial average even with periodic boundary conditions.

The Kuramoto-Sivashinsky equation $u_t+u{u}_x+u_{xx}+u_{xxxx}=0$ combines destabilizing long-wave growth, stabilizing fourth-order diffusion, and nonlinear transport. Its potential form is ${\Phi }_t=-{\Phi }_{xx}-{\Phi }_{xxxx}+({\Phi }_x{)}^2$, with $u=-2{\Phi }_x$. It arises as a long-wave phase amplitude equation near loss of the Benjamin-Feir stability condition. On a domain of period $P$, the zero solution's Fourier modes grow at rate $k^2-k^4$, $k=2\pi n/P$.

Fix zero mean and period $P$ in the Kuramoto-Sivashinsky equation. At $P=2\pi$ the first Fourier mode becomes neutral and all higher modes are damped. For $k=2\pi /P$, expand $u=a\sin (kx)+b\sin (2kx)+\cdots$. The first equations are $\dot{a}={\lambda }_{1}a+kab/2+\cdots$, $\dot{b}={\lambda }_{2}b-k{a}^2/2+\cdots$, with ${\lambda }_j=(jk{)}^2-(jk{)}^4$. Slaving the second harmonic gives $\dot{a}={\lambda }_{1}a+k^2{a}^3/(4{\lambda }_2)+\cdots$. Since ${\lambda }_2=-12$ at onset, this is a supercritical bifurcation. The small nonzero branch is stable in the sense of orbital stability against disturbances of the same fixed period, modulo translation; stability against disturbances of larger period does not follow from this local argument.

Green's identities relate volume integrals involving gradients and Laplacians to boundary values and normal derivatives. They follow from the divergence theorem and include Green's first identity, Green second identity and Green's third identity.

Green's third identity represents a function inside a domain through its boundary values, normal derivatives, and a volume term involving its Laplacian.

For smooth functions $u,v$ on a region $V$,

If $f$ is constant on the boundary and $g$ is a regular harmonic function throughout a volume, the divergence theorem applied to $f\nabla g$ gives ${\int }_V\nabla f\cdot \nabla g={\int }_{\partial V}f{\partial }_{n}g$. The constant boundary value factors out; a second application of the divergence theorem makes the remaining flux ${\int }_V\Delta g=0$. This requires enough regularity to use Green's first identity over the entire volume.

Although $1/r$ is harmonic away from the origin in three dimensions, it is not a regular harmonic function on a ball containing the origin. Its gradient is $-\hat{r}/r^2$, so the displayed improper volume integral follows by canceling the $r^2$ in the spherical volume element. On the annulus $\epsilon <r<a$, Green's first identity gives outer boundary contribution $-4\pi$ and inner contribution $4\pi \epsilon /a$, recovering the same limit. Its global Laplacian contains a Dirac delta distribution, so the regular-harmonic gradient-pairing theorem is inapplicable.

A maximum principle bounds a solution through its boundary or initial data, often forbidding nonconstant solutions from attaining an interior extremum. Weak and strong versions depend on the differential operator and hypotheses. The maximum principle for harmonic functions and its strong counterpart are examples.

For a classical solution of $u_t-\Delta u+cu=0$ with $c\ge 0$, the positive part of its maximum is bounded by the corresponding maximum on the initial and lateral parabolic boundary. When $c>0$, a positive maximum in space at an interior or final time gives $u_t\ge 0$, $\Delta u\le 0$, and $u_t-\Delta u+cu>0$, a contradiction. For $c=0$, a small strictly increasing exponential time perturbation gives the weak inequality by a limiting argument. Applied to both signs of the difference of two solutions, the principle proves uniqueness for fixed initial and boundary data.

On $x,t>0$, the displayed smooth nonzero function satisfies $w_t=x{w}_{xx}$. For each fixed $t>0$ it tends to zero at $x=0$ and at spatial infinity; for each fixed $x>0$ it tends to zero as $t\downarrow 0$. Thus these three pointwise limits do not determine a unique solution: a multiple of $w$ may be added to any particular solution of the same forced equation.

To verify the PDE, set $\eta =x/t$. Both $w_t$ and $x{w}_{xx}$ equal $\eta (\eta -2)e^{-\eta }/t^2$. The missing control is at the corner: $w(t,t)=1/(et)$ diverges as $t\downarrow 0$. There is no contradiction with the weak parabolic maximum principle, because a supremum on a past strip may be infinite or unattained. A bounded-solution class excludes this example. A standard uniqueness proof there applies the maximum principle to the difference minus $\delta (1+x+t)$ and lets $\delta \downarrow 0$. Conditions at infinity and at initial time must control the relevant strip, not only each fixed-variable limit.

Let $\Omega$ be a bounded domain, with $u\in C^2(\Omega )\cap C(\overline{\Omega })$ real valued and $c>0$. At a positive interior maximum, $\Delta u\le 0$ but $cu>0$, contradicting the equation. A negative interior minimum gives the opposite contradiction. Thus any value with absolute magnitude exceeding the boundary maximum would be impossible, proving the displayed bound. In particular zero boundary data force $u=0$, and subtracting two solutions proves uniqueness for the corresponding inhomogeneous Dirichlet problem.

Let $\Omega$ be a connected domain and $u$ a classical subsolution ofwith uniformly positive definite diffusion matrix, continuous bounded coefficients, and $c\ge 0$. If $u\le 0$ on $\Omega \times (0,T]$ and $u(x_0,t_0)=0$ at a spatial interior point with $t_0>0$, then $u=0$ on $\Omega \times (0,t_0]$. Continuity supplies the same conclusion at initial and lateral points where the solution extends continuously. There is generally no conclusion about times after $t_0$ for a subsolution.

One proof applies the strict positivity principle to $w=-u\ge 0$, a supersolution: any positive value at an earlier interior point propagates to positive values at every later interior point along a chain of overlapping interior balls, contradicting $w(x_0,t_0)=0$. Nonnegative zeroth-order terms can be bounded on each compact time cylinder and removed from this positivity argument by an exponential multiplier. For the inequality $u_t-\Delta u+b\cdot \nabla u+u^3\le 0$, use $c=u^2\ge 0$ after the weak maximum principle has established $u\le 0$.

If a harmonic function on a connected domain attains an interior maximum or minimum, it is constant. The mean value property forces equality throughout every sufficiently small ball around an interior extremum, and connectedness propagates the equality.

This is the strong harmonic instance of the maximum principle.

A continuous function harmonic on a bounded domain attains its maximum and minimum on the boundary. Apply the strictly subharmonic result to $u+ϵ|x{|}^2$ and let $ϵ\downarrow 0$, then apply the same argument to $-u$.

If $u,v$ are harmonic functions on a bounded domain and continuous on its closure, their difference is a harmonic function. Its minimum is attained on the boundary by the maximum principle for harmonic functions applied to its negative. Nonnegative boundary difference therefore implies nonnegative interior difference. Applying the comparison in both directions proves uniqueness for the Dirichlet problem whenever a classical solution exists. This is an order statement and does not itself establish existence.

Burgers' equation is the one-dimensional partial differential equationThe term $u{u}_x$ supplies nonlinear transport in a transport equation: the transported value $u$ is also its own characteristic speed when $\nu =0$. Larger values catch up with smaller values ahead of them, steepening a decreasing profile. The diffusion term $\nu u_{xx}$ opposes this steepening. Positive $\nu$ gives the viscous Burgers equation; setting $\nu =0$ gives the Inviscid Burgers equation.

This small equation makes three different behaviors accessible in one model: Burgers characteristic breaking from continuous decreasing data, an entropy shock for a downward step, and a rarefaction wave for an upward step. The Burgers Riemann problem with positive flux and the Viscous Burgers travelling shock below give exact examples. These are mathematical calculations with chosen data; the figures contain no experimental measurements.

Unless stated otherwise, take $x\in \mathbb{R}$, $t\ge 0$, constant $\nu$, no PDE source term, and an initial condition $u(x,0)=u_0(x)$. There are no finite endpoints requiring boundary conditions. Where a profile has different limiting states at $x=\pm \infty$, those states are specified explicitly. On a finite interval, boundary conditions become part of the problem: positive diffusion normally requires a condition at each endpoint, whereas the Inviscid Burgers equation requires appropriate inflow transport boundary conditions. With dimensional variables, $u$ has units of velocity and $\nu$ of kinematic viscosity; all numerical examples below use dimensionless quantities.

The Burgers' equation with constant linear damping has the displayed form. For zero viscosity and $\gamma >0$, characteristics are $u=u_0(a)e^{-\gamma t}$ and $x=a+u_0(a)(1-e^{-\gamma t})/\gamma$. A negative initial slope can make this coordinate map cease to be invertible, producing shock formation by characteristic intersection. The transformation $u=e^{-\gamma t}v$, $\tau =(1-e^{-\gamma t})/\gamma$ converts the inviscid equation to the ordinary inviscid Burgers' equation. Across a thin viscous layer the damping contributes only at lower order to the leading Rankine-Hugoniot condition.

The viscous Burgers equation combines a nonlinear transport equation with diffusion with constant $\nu >0$. On the infinite line, a bounded initial condition gives a smooth function solving the equation for every positive time; a jump in the initial condition is smoothed rather than retained as a discontinuity. Small $\nu$ can nevertheless produce a steep layer, as the Viscous Burgers travelling shock illustrates. The Cole-Hopf transformation $u=-2\nu {\partial }_x\log \varphi$, with positive $\varphi$, converts this equation to the heat equation ${\varphi }_t=\nu {\varphi }_{xx}$.^[ref]

Positive diffusion also explains the admissibility of the inviscid limit. For the entropy flux for a scalar conservation law associated with $\eta (u)=u^2/2$, direct multiplication by $u$ givesThe last term is nonpositive. With convergence sufficient to pass to the limit, the vanishing viscosity approximation therefore retains an entropy inequality rather than just the weak solution equation. For general bounded data, the full Kruzhkov entropy inequalities specify the entropy solution; the quadratic identity is a useful illustration of dissipation.^[ref]

A travelling wave of the viscous Burgers equation can connect constant states $u_L>u_R$ through a smooth decreasing layer. Require $u(x,t)=U(z)$, $z=x-st$, $U(-\infty )=u_L$, $U(+\infty )=u_R$, and $U^{\prime }\to 0$ at both ends. Substitution and one integration giveThe speed follows by evaluating the integration constant at both end states. It is precisely the Rankine-Hugoniot condition for conservation law flux $f(u)=u^2/2$. Choosing the center $U(0)=s$ gives the displayed hyperbolic tangent profile. Its characteristic width scales as $\nu /(u_L-u_R)$; a smaller positive $\nu$ makes a thinner smooth layer.

For an original worked example in dimensionless quantities, take $u_L=2$, $u_R=0$ and $s=1$:This is an exact classical solution on $\mathbb{R}$ with its own smooth initial condition $u(x,0)=1-\tanh (x/(2\nu ))$ and limiting states $2,0$. It is a family of moving profiles, not the finite-$\nu$ solution starting from an exact discontinuous step. For that different initial condition, the Cole-Hopf transformation supplies the evolution. At $\nu =1/2$, this front starts from the same $1-\tanh x$ profile as the Burgers characteristic breaking example: positive diffusion maintains a smooth moving layer while the inviscid profile breaks at $t=1$.

To check the partial differential equation explicitly, set $w=(x-t)/(2\nu )$ and $S=1-{\tanh }^{2}w$. ThenThe independent Rankine-Hugoniot condition check gives $s=(0-2)/(0-2)=1$, using $f(2)=2$ and $f(0)=0$. At fixed $x\ne t$, the vanishing viscosity approximation tends to the entropy shock $2$ for $x<t$ and $0$ for $x>t$. At the center the smooth profiles stay at $1$; changing the value on the discontinuity does not change the limiting weak solution.

For $u_L<u_R$, the same integrated ordinary differential equation has $U^{\prime }<0$ between the two states when $\nu >0$, so it cannot supply an increasing connecting front. The inviscid admissible expansion is the rarefaction wave of the Burgers Riemann problem with positive flux.

The cubic-flux variant replaces the quadratic conservative flux of Burgers' equation by $-q^3/3$. Its inviscid characteristic speed is $-q^2$, while positive diffusivity regularizes shock waves. It is not linearized by the usual Cole-Hopf transformation.

For a front written in $s=Z+c\theta$ connecting $0$ at $s=-\infty$ to nonzero $\beta$ at $s=+\infty$, one integration gives $ϵc^2{q}^{\prime }=q-c{q}^3/3$. The endpoint condition fixes $c=3/{\beta }^2$. Setting $w=q^2$ gives a logistic equation and the displayed profile, with the sign of $q$ selected by $\beta$. The propagation speed in the $\theta$ coordinate is $-{\beta }^2/3$, agreeing with the Rankine-Hugoniot condition for flux $-q^3/3$.

The Fay solution is a periodic solution of the negative-flux viscous Burgers equation for $Z>0$. Its Fourier series has exponentially decreasing coefficients at each positive $\delta Z$. Expanding $1/\sinh (ns)=2{\sum }_{j\ge 0}{e}^{-(2j+1)ns}$ and summing the sine series givesFor $s\gg 1$, the first harmonic gives $q\sim 4\delta e^{-s}\sin \theta$. The Fay shock-layer asymptotics describe its opposite, small-$s$ regime.

For $|\theta |\ll 1$ and $s=\delta Z\ll 1$, the Fay solution becomes $q\sim 4\delta \theta {\sum }_{j\ge 0}[((2j+1)s{)}^2+{\theta }^2{]}^{-1}$. The partial-fraction identity for the hyperbolic tangent sums this to the displayed smooth shock layer. It resolves the odd transition from $-\pi /Z$ to $+\pi /Z$ on angular thickness $O(\delta Z)$; at $\theta =0$ both sides vanish and the comparison is read through their derivatives.

A localized negative-flux Burgers N-wave has initial profile $-U\theta$ on $|\theta |<L$ and zero outside, for $U,L>0$. The Cole-Hopf solution for a Burgers N-wave expresses its viscous evolution in Gaussian interval masses. In the vanishing-viscosity limit, the interior slope becomes $-U/(1+UZ)$ and its entropy-shock fronts move to $\pm L\sqrt{1+UZ}$.

For the negative-flux viscous Burgers equation, the Cole-Hopf transformation uses $f=2\alpha {\partial }_{\theta }\log \psi$. Initial N-wave data give ${\psi }_0=\exp [U(L^2-{\theta }^2)/(4\alpha )]$ inside $[-L,L]$ and one outside. Completing the square in the heat kernel convolution gives, with $a=1+UZ$,The Gaussian interval masses and their exponentially weighted tails must be treated together when taking the small-diffusion limit.

For positive diffusivity, the Cole-Hopf transformation with $f=2\alpha {\partial }_{\theta }\log \psi$ converts the negative-flux equation into the heat equation. Step data give a ratio of two Gaussian-tail integrals $J$. At $\theta =-UZ/2$, $J=1$ and $f=U/2$. Its vanishing-viscosity limit selects the shock or rarefaction wave in the Burgers Riemann problem with negative flux, depending on the sign of $U$.

The viscous Burgers step solution with negative flux is symmetric around $\theta =-Uz/2$ after subtracting $U/2$. Its Gaussian-tail ratio equals one precisely at this midpoint, giving $f=U/2$ for either sign of $U$. For $U>0$ this point becomes the entropy-shock center in the vanishing viscosity approximation; for $U<0$ it is the center of a rarefaction wave. The same trajectory therefore does not by itself identify a shock.

The Cole-Hopf transformation $u=-2\nu {\varphi }_x/\varphi$ converts the viscous Burgers equation into the heat equation ${\varphi }_t=\nu {\varphi }_{xx}$. Initial data $u(x,0)=g(x)$ correspond to $\varphi (x,0)=C\exp [-(2\nu {)}^{-1}{\int }^{x}g(s)ds]$.

Substituting $u=-2\nu {\partial }_x\log \Psi$ into the viscous Burgers equation gives ${\partial }_x[({\Psi }_t-\nu {\Psi }_{xx})/\Psi ]=0$. Integration permits the arbitrary function $C(t)$ in the title. Multiplication of $\Psi$ by $e^{-\int C(t)dt}$ preserves $u$ and reduces its equation to the heat equation. Thus the heat-equation representation uses a permissible time normalization rather than a condition uniquely imposed on every representative $\Psi$.

For $q_Z-q{q}_{\theta }=\delta q_{\theta \theta }$ and initial data $q=\sin \theta$, the Cole-Hopf transformation $q=2\delta {\psi }_{\theta }/\psi$ gives $\psi (\theta ,0)=\exp [-\cos \theta /(2\delta )]$. The Fourier series of this exponential and the heat equation give the displayed exact positive diffusion solution. Its first harmonic gives $q\sim 4\delta [I_1(a)/I_0(a)]e^{-\delta Z}\sin \theta$. The large-argument asymptotic expansion of a modified Bessel function makes the ratio tend to one as $\delta \to 0$.

For the negative-flux viscous Burgers equation $q_Z-q{q}_{\theta }=ϵq_{\theta \theta }$, set $q=2ϵ{\partial }_{\theta }\log \psi$. A sum of exponentials in the initial heat data evolves into $\psi ={\sum }_j{e}^{A_j}$, giving the displayed softmax function average. Two terms produce a traveling tanh shock from the lower to the higher state, with speed minus their mean.

For ordered states $q_1<\cdots <q_N$, in the moving frame $\theta =sZ+\xi$ the first and last exponents in the finite-exponential Cole-Hopf solution have the same leading coefficient. Each interior state's coefficient is smaller by $(q_j-q_1)(q_N-q_j)/(4ϵ)>0$. Thus the large-$Z$ moving-frame profile is the two-state shock between the extremes. At fixed $\theta$, the endpoint of largest absolute value dominates; equal opposite extremes leave a stationary shock.

A semilinear partial differential equation is linear in its highest-order derivatives, with their coefficients depending only on the independent variables; lower-order terms may depend nonlinearly on the solution and its lower derivatives. A semilinear heat equation such as $u_t=\Delta u+f(u)$ is an example. This differs from a quasilinear equation, whose highest-order derivative coefficients can depend on the solution or its lower derivatives.

A heat equation with a nonlinear reaction term depending on the value of its solution.

An obstacle constraint requires a solution to lie above a given function. In an optimal stopping problem, the value dominates the payoff, satisfies a generator equation where continuation is optimal, and a generator inequality where stopping is optimal. The complementary conditions can be expressed as a maximum equal to zero.

The noncontact set is where the unknown function is strictly above its obstacle. For continuous functions it is open. Every smooth variation with compact support in this set remains feasible with either sign when its amplitude is small enough. The first variation therefore vanishes there, yielding the unconstrained Euler-Lagrange equation. On the complementary contact set of an obstacle problem, only the appropriate one-sided variational inequality is available globally.

The contact set is the region where the obstacle inequality $w\ge f$ is active. On its complement the obstacle supplies no reaction; its boundary is a free boundary, determined along with the unknown solution. A point can belong to the contact set while its contact force is zero, as for a taut horizontal string just touching an obstacle at one point. Thus equality of the gap alone does not imply strictly positive pressure.

For a string of positive tension $T$ with fixed endpoints $w(\pm 1)=0$, zero transverse loading and obstacle $f=h-x^2$, $0\le h\le 1$, equilibrium minimizes $T{\int }_{-1}^1(w^{\prime }{)}^2/2dx$ among functions above $f$. On a free interval $w^{\prime \prime }=0$, while in contact $w=f$. Smooth tangency of the right line through $(1,0)$ to the parabola at $x=s$ requires $h-s^2=2s(1-s)$, yielding the displayed root in $[0,1]$.

The two pieces join with equal value and slope. On the free portions $w-f=(|x|-s{)}^2\ge 0$; in contact $-T{w}^{\prime \prime }=2T$ is the nonnegative reaction. These facts verify the complementarity conditions. The associated variational inequality and the exact energy difference prove global minimality; strict convexity with fixed endpoints proves uniqueness. At $h=0$ contact reduces to one point; at $h=1$ it fills the whole interval.

Minimizing the log-cosh energy over zero-trace functions above an obstacle yields the displayed inequality for every feasible competitor. Convexity supplies the one-sided admissible segments. On the open noncontact region, compactly supported variations of both signs are feasible because the gap from the obstacle has a positive minimum on their support. Their first variations vanish, giving $-\Delta u+\tanh u=0$ there. This stationarity equation need not hold on the contact set.

A first-contact argument compares a smooth solution with a subsolution or supersolution. At a first spatial minimum of their difference, the Hessian is nonnegative; a strictly favorable time derivative then contradicts contact. For $u_t=u^2{u}_{xx}+u^3$, a positive decreasing constant-in-space barrier proves that positive periodic initial data stay positive for the lifetime of the smooth solution.

For $u_t=\Delta u+\lambda u-u^3$ with zero Dirichlet boundary condition, choose $K\ge \max (\Vert u_0{\Vert }_{\infty },\sqrt{\max (\lambda ,0)})$. The constants $K$ and $-K$ are a supersolution and a subsolution. More explicitly, testing with $(u-K{)}_{+}$ gives $\frac{1}{2}\frac{d}{dt}\Vert (u-K{)}_{+}{\Vert }_2^2\le \lambda \Vert (u-K{)}_{+}{\Vert }_2^2$, since the reaction derivative is at most $\lambda$ and the reaction at $K$ is nonpositive. The Gronwall inequality and the corresponding argument for $-u$ prove that $[-K,K]$ is invariant. A local variation-of-constants formula in the supremum norm then has a lifespan depending only on $K$, which prevents finite-time loss of the classical solution.

The domain of dependence of a solution value consists of the initial and boundary locations whose data can influence it. For $u_t+a{u}_x=0$ on the whole line, $u(x,t)=u(x-at,0)$, so its initial-data domain of dependence is the foot of the characteristic. For a finite-difference scheme, repeated stencil dependencies determine a numerical domain of dependence. The Courant–Friedrichs–Lewy condition requires the numerical domain to cover the physical one for convergence; it is necessary under its usual hyperbolic hypotheses, but not in general sufficient for stability.

For Noncharacteristic Cauchy data parameterized by $t\ge 0$, the method of characteristics produces a surface $\Phi (s,t)=(x,y,z)$. The part uniquely fixed as a classical graph through the data must retain both the allowed footpoint range and a nonsingular branch of the projected map. On a branch where inversion gives $(s(x,y),t(x,y))$, require $t(x,y)\ge 0$, together with the branch condition and a nonzero Jacobian determinant. The strict inequality $t>0$ describes its open interior; the endpoint characteristic $t=0$ supplies a boundary trace. A fold caustic can supply another boundary.

An algebraic formula obtained by eliminating $s,t$ can extend beyond this domain of dependence. Its values there need not be selected by the original data, because their characteristic footpoints can lie on the unprescribed half of the data curve. Tracking the parameter range is therefore part of the uniqueness argument, separately from selecting a real square-root branch.

For the one-dimensional wave equation $U_{ss}=U_{rr}$ with Cauchy data on $s=s_0$, $a\le r\le b$, the solution is determined where both characteristic footpoints $r\pm (s-s_0)$ lie in $[a,b]$. This is the diamond displayed in the title, or $|r-(a+b)/2|+|s-s_0|\le (b-a)/2$. In characteristic coordinates, it is a rectangle of known arguments for the two travelling-wave functions. A smooth invertible coordinate transformation maps it to a region bounded by the original equation's characteristic curves; it need not remain a Euclidean diamond. Both forward and backward determination can be described this way. Outside the region, changing an unspecified part of one travelling-wave function can change the solution without changing the supplied finite data segment.

For $r=\sqrt{x}$, $s=y^{3/2}/3$ and data on $y=1$, $1/4\le x\le 1$, the transformed segment is $s_0=1/3$, $a=1/2$, $b=1$. The known characteristic arguments lie in $[5/6,4/3]$ and $[1/6,2/3]$. The corresponding original region has curved boundaries despite being a rectangle in characteristic coordinates and a diamond in $(r,s)$ coordinates.

A linear partial differential equation is a partial differential equation in which the unknown function and its partial derivatives occur linearly, with coefficients independent of the unknown. Its homogeneous solutions obey the superposition principle.

For example, the heat equation permits adding independently decaying Fourier modes. For an equation $Lu=f$ with a fixed nonzero PDE source term, adding two solutions gives $L(u+v)=2f$, so the sum generally solves a different problem. Likewise, adding solutions adds their initial conditions and boundary conditions.

For a second-order real linear partial differential equation in two variables, an invertible local change of variables simplifies its principal part. At regular points, the hyperbolic case has mixed-derivative principal part $U_{\xi \eta }$, the elliptic case has principal part $U_{\xi \xi }+U_{\eta \eta }$, and the parabolic case has rank-one principal part $U_{\xi \xi }$, after multiplication by nonzero factors. Lower-order terms and forcing must also be transformed with the chain rule; simplifying only the principal part does not in general give the full canonical equation. In the hyperbolic partial differential equation case the new variables can be chosen constant along the two characteristic families.

For a second-order hyperbolic partial differential equation, the usual reduction to a mixed derivative uses twice continuously differentiable characteristic coordinates. Continuity of the principal coefficients alone does not ensure this regularity, even locally. Consider $R=0$, $S=1/2$ and $T=|x|$ near the origin; the discriminant $S^2-RT$ is the positive constant $1/4$. A characteristic coordinate must satisfyThe two characteristic families have first integrals $x$ and $y-x|x|/2$. Locally, any coordinate transverse to the first family has the form $\eta =h(y-x|x|/2)$ with $h^{\prime }\ne 0$. If this were twice continuously differentiable, its restriction at $x=0$ would make $h$ twice continuously differentiable. But its second $x$ derivative approaches $-h^{\prime }(y)$ from the right and $h^{\prime }(y)$ from the left, a contradiction. Independent characteristic coordinates cannot both choose the other family. Thus no twice continuously differentiable characteristic coordinate system exists near the origin.

With sufficiently smooth coefficients, the two distinct characteristic direction fields do yield smooth local coordinates. If $J$ is their Jacobian determinant and $B$ is the transformed mixed coefficient, then $B^2=J^2(S^2-RT)>0$. Division by $2B$ gives the mixed-derivative canonical form, including all lower-order terms produced by the chain rule. This is a local result; global intersections of characteristic curves can create separate obstructions.

For $\xi =x+e^y$, $\eta =x-e^y$, the Jacobian determinant is $-2e^y$ and the transformed domain is $\xi >\eta$. The chain rule gives $u_{yy}=e^{2y}(U_{\xi \xi }-2U_{\xi \eta }+U_{\eta \eta })+e^y(U_{\xi }-U_{\eta })$, while $u_y=e^y(U_{\xi }-U_{\eta })$. The drift terms cancel, leaving $U_{\xi \eta }=0$. Thus the general solution is $F(x+e^y)+G(x-e^y)$; its characteristic curves are the level sets of the two new coordinates.

An ill-posed problem fails existence, uniqueness or continuous dependence in the specified function spaces. Backward heat equation evolution is unstable because its Fourier modes grow like $e^{t|\xi {|}^2}$, making arbitrarily small high-frequency errors grow without a uniform bound.

A linear partial differential operator sends a function to a linear combination of its partial derivatives. If its coefficients are constant, the Fourier transform of a derivative turns it into multiplication by its polynomial symbol $P(\xi )$.

The elliptic Liouville equation $\Delta u=e^{2u}$ describes conformal metrics of constant negative Gaussian curvature under a common sign convention.

A radial function depends only on the distance from the origin: $f(x)=g(|x|)$. Differential operators then reduce to radial formulas such as $\Delta f=g^{\prime \prime }+(n-1)g^{\prime }/r$.