Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 71 / 4

The Fourier sine basis of the unit square is $e_{j\ell }(x,y)=2\sin (j\pi x)\sin (\ell \pi y)$, $j,\ell \ge 1$, normalized in the L2 inner product. It satisfies the zero Dirichlet boundary conditions and $\Delta e_{j\ell }=-{\pi }^2(j^2+{\ell }^2)e_{j\ell }$. Expand the given initial datum as $u_0=\sum a_{j\ell }{e}_{j\ell }$, where $\sum |a_{j\ell }{|}^2=\Vert u_0{\Vert }_2^2$. The solution isFor $\kappa \le 2{\pi }^2$, every exponent is nonpositive. The series converges in $L^2$ for $t\ge 0$, tends to $u_0$ in $L^2$ as $t\downarrow 0$ by dominated convergence, and differentiates termwise for $t>0$ because the Gaussian high-frequency decay dominates every polynomial derivative factor. This constructs a classical positive-time solution and the appropriate mild solution at the possibly rough initial time.

Its norm and the difference of two such solutions satisfyExistence, uniqueness in the natural $L^2$ solution class, and continuous dependence follow. For uniqueness one can project any weak solution onto each sine mode, obtaining the same scalar linear equation and initial coefficient. Thus this is a Hadamard well-posed problem, with a contractive strongly continuous semigroup in the stated range.

To prove pointwise decay from merely $L^2$ data, use $|e_{j\ell }|\le 2$ and the Cauchy-Schwarz inequality:For $t\ge 1$, factor out $e^{2(\kappa -2{\pi }^2)t}$; the remaining sum is bounded by ${\sum }_{j,\ell \ge 1}{e}^{-2{\pi }^2(j^2+{\ell }^2-2)}<\infty$. ThereforeAt equality the first sine mode is stationary, so strict inequality is essential for decay of every datum. This is the Dirichlet heat-reaction threshold on the unit square. The equation is actually well posed on every fixed finite time interval for any fixed real $\kappa$; beyond the threshold it has a growing first mode and loses long-time contractivity. The restricted well-posedness assertion in the question is therefore sufficient rather than necessary.

Take $M$ interior points in each direction and spacing $h=1/(M+1)$. The one-dimensional Dirichlet discrete Laplacian has sine eigenvectors $v_i^{(j)}=\sin (j\pi ih)$, becauseThe five-point Dirichlet Laplacian as a Kronecker sum therefore has product eigenvectors $v_i^{(j)}{v}_r^{(\ell )}$. Adding the reaction term gives modal growth ratesThe semidiscrete matrix is real symmetric, so these modes are orthogonal in the mesh-weighted discrete L2 norm. Non-growth for all times is equivalent to every rate being nonpositive. The largest is ${\lambda }_{11}$, giving the five-point heat-reaction stability thresholdAt equality the first grid mode is neutral; above it that mode grows exponentially. This criterion concerns the spatially semidiscrete, continuous-time evolution. Any subsequent time integrator has an additional stability requirement.

The threshold expands as $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$ and is below the continuum threshold for every finite mesh. Thus at the continuum neutral value the grid first mode has a small positive $O(h^2)$ rate. This is compatible with finite-time convergence, although it prevents an all-time contraction on that fixed mesh. In the weaker finite-time mesh-uniform sense, all fixed $\kappa$ have the bound $\Vert U(t){\Vert }_h\le e^{\kappa t}\Vert U(0){\Vert }_h$ because the discrete Laplacian is nonpositive. The boxed range is the natural non-growing stability range, matching the threshold comparison in the preceding part; finite-time stability versus power boundedness is a distinct question.