The theory of [second-order elliptic equations](/page/Second-Order%20Elliptic%20Equations) concerns equilibrium states — solutions to $Lu = f$ that do not change in time. In many physical systems, however, the quantity of interest (temperature, concentration, probability density) evolves toward equilibrium, and the transient behaviour is itself the object of study. This leads to **parabolic equations**, where a time derivative is coupled to an elliptic spatial operator. The archetype is the [heat equation](/page/Heat%20Equation) $u_t - \Delta u = 0$: the temperature $u(x,t)$ at position $x$ and time $t$ evolves by diffusion until it reaches the steady state $\Delta u = 0$. More generally, a second-order parabolic equation takes the form \begin{align*} u_t + Lu = f \quad \text{in } U_T, \end{align*} where $L$ is the elliptic operator from the [elliptic theory](/page/Second-Order%20Elliptic%20Equations) acting on the spatial variables, and $U_T := U \times (0, T]$ is the **parabolic cylinder**. The sign convention $u_t + Lu = f$ (rather than $u_t - Lu$) is chosen so that $L = -\Delta$ gives the standard heat equation $u_t - \Delta u = f$. The analytic structure of parabolic equations differs fundamentally from the elliptic case. An elliptic equation is an infinite-dimensional analogue of a linear system $Ax = b$; a parabolic equation is an infinite-dimensional analogue of an initial value problem $\dot{x} = Ax + b$, $x(0) = x_0$. The solution at time $t$ depends on the entire history $[0, t]$, and the natural function spaces must account for the asymmetry between the spatial and temporal variables. ## Setup and Notation Throughout, we fix: - A bounded [open set](/page/Open%20Set) $U \subseteq \mathbb{R}^n$ with $C^1$ [boundary](/page/Boundary) $\partial U$. - A time horizon $T > 0$. - The **parabolic cylinder** $U_T := U \times (0, T]$. - The **parabolic boundary** $\Gamma_T := \overline{U_T} \setminus U_T = (\partial U \times [0, T]) \cup (U \times \{0\})$. The parabolic boundary $\Gamma_T$ consists of the points from which information "flows into" the cylinder: the spatial boundary $\partial U \times [0, T]$ (where boundary conditions are imposed) and the initial slice $U \times \{0\}$ (where the initial condition is prescribed). Unlike the elliptic case, where data is imposed on the entire boundary $\partial U$, the "top" of the cylinder $U \times \{T\}$ is **not** part of $\Gamma_T$ — the solution there is determined by the equation.  We consider the general second-order operator in divergence form, exactly as in the [elliptic setting](/page/Second-Order%20Elliptic%20Equations): [definition: Parabolic Operator] Let $L$ be a second-order elliptic operator in divergence form with time-dependent coefficients: \begin{align*} Lu(x, t) := -\sum_{i,j=1}^n \partial_{x_j}\bigl(a_{ij}(x,t)\,\partial_{x_i} u\bigr) + \sum_{i=1}^n b_i(x,t)\,\partial_{x_i} u + c(x,t)\,u, \end{align*} where $a_{ij}, b_i, c \in L^\infty(U_T)$. The operator $L$ acts only on the spatial variables $x$; the time variable $t$ enters as a parameter. The associated **parabolic equation** with source $f$ and initial-boundary data is the system: \begin{align*} \begin{cases} u_t + Lu = f & \text{in } U_T, \\ u = 0 & \text{on } \partial U \times [0, T], \\ u = g & \text{on } U \times \{0\}. \end{cases} \end{align*} [/definition] [definition: Uniform Parabolicity] The operator $\partial_t + L$ is **uniformly parabolic** if $L$ is uniformly elliptic in the spatial variables, uniformly in time: there exists $\theta > 0$ such that \begin{align*} \sum_{i,j=1}^n a_{ij}(x,t)\,\xi_i\xi_j \ge \theta\,|\xi|^2 \quad \text{for a.e. } (x,t) \in U_T \text{ and all } \xi \in \mathbb{R}^n. \end{align*} [/definition] ## Function Spaces for Evolution Equations The fundamental new difficulty in parabolic theory is that the solution $u(\cdot, t)$ lives in a spatial [Sobolev space](/page/Sobolev%20Space) for each fixed time $t$, and we must control how $u$ varies as $t$ changes. This requires **Bochner spaces** — the natural $L^p$ spaces for [functions](/page/Function) taking values in a [Banach space](/page/Banach%20Space). [definition: Bochner Space $L^2(0,T; X)$] Let $X$ be a Banach space with norm $\|\cdot\|_X$. The space $L^2(0, T; X)$ consists of all (equivalence classes of) strongly measurable functions $u: [0, T] \to X$ such that: \begin{align*} \|u\|_{L^2(0,T;X)} := \left(\int_0^T \|u(t)\|_X^2\,dt\right)^{1/2} < \infty. \end{align*} [/definition] The solution space for parabolic equations is built from two Bochner spaces with different roles. The space $L^2(0, T; H^1_0(U))$ controls the **spatial regularity** (one spatial derivative in $L^2$) at each time, while $L^2(0, T; H^{-1}(U))$ is where the time derivative $u_t$ lives — it need not be an $L^2$ function, only a distribution in space. [definition: The Parabolic Solution Space] Define the space: \begin{align*} \mathcal{W} := \bigl\{u \in L^2(0, T; H^1_0(U)) : u_t \in L^2(0, T; H^{-1}(U))\bigr\}, \end{align*} equipped with the norm: \begin{align*} \|u\|_{\mathcal{W}} := \|u\|_{L^2(0,T;H^1_0(U))} + \|u_t\|_{L^2(0,T;H^{-1}(U))}. \end{align*} Here $u_t$ denotes the [distributional](/page/Distribution) time derivative: $u_t \in L^2(0, T; H^{-1}(U))$ means that the map $t \mapsto u_t(t) \in H^{-1}(U)$ is $L^2$-[integrable](/page/Integral), and for every $v \in H^1_0(U)$: \begin{align*} \int_0^T (u_t \circ v)\,\varphi(t)\,dt = -\int_0^T (u \circ v)\,\varphi'(t)\,dt \quad \text{for all } \varphi \in C_c^\infty(0, T), \end{align*} where $\circ$ denotes the duality pairing between $H^{-1}(U)$ and $H^1_0(U)$, as in the [elliptic theory](/page/Second-Order%20Elliptic%20Equations). [/definition] [remark: Continuity in Time] A crucial property of the space $\mathcal{W}$ is the embedding $\mathcal{W} \hookrightarrow C([0, T]; L^2(U))$. That is, every $u \in \mathcal{W}$, after possible modification on a [set](/page/Set) of measure zero in time, is a continuous function from $[0, T]$ into $L^2(U)$. This is essential: it ensures that the initial condition $u(\cdot, 0) = g$ is meaningful as an equality in $L^2(U)$. The proof uses the fact that the triple $H^1_0(U) \subset L^2(U) \subset H^{-1}(U)$ forms an **evolution triple** (or Gelfand triple), and the embedding theorem for functions with values in such triples gives the desired temporal [continuity](/page/Continuity). [/remark] ## Weak Solutions As in the elliptic case, we formulate the PDE in weak form by testing against spatial [test functions](/page/Test%20Function). The key difference is that the weak equation must hold at almost every time $t$, rather than as a single static identity. [definition: Weak Solution of the Parabolic Equation] Let $f \in L^2(0, T; H^{-1}(U))$ and $g \in L^2(U)$. A function $u \in \mathcal{W}$ is a **weak solution** of the initial-boundary value problem \begin{align*} \begin{cases} u_t + Lu = f & \text{in } U_T, \\ u = 0 & \text{on } \partial U \times [0, T], \\ u(\cdot, 0) = g & \text{in } U, \end{cases} \end{align*} if the following two conditions hold: 1. **Weak PDE.** For every $v \in H^1_0(U)$ and a.e. $t \in (0, T)$: \begin{align*} u_t(t) \circ v + B[u(t), v; t] = f(t) \circ v, \end{align*} where $B[\cdot, \cdot\,; t]$ is the bilinear form associated with $L$ at time $t$: \begin{align*} B[u, v; t] := \int_U \biggl(\sum_{i,j=1}^n a_{ij}(\cdot, t)\,\partial_{x_i} u\,\partial_{x_j} v + \sum_{i=1}^n b_i(\cdot, t)\,(\partial_{x_i} u)\,v + c(\cdot, t)\,uv\biggr)\,d\mathcal{L}^n. \end{align*} 2. **Initial condition.** $u(\cdot, 0) = g$ in $L^2(U)$. [/definition] The key structural difference from the [elliptic weak formulation](/page/Second-Order%20Elliptic%20Equations) is the term $u_t(t) \circ v$: the time derivative acts as a forcing in $H^{-1}(U)$ at each instant, coupling the spatial problem to the temporal evolution. Condition (2) is meaningful by the embedding $\mathcal{W} \hookrightarrow C([0,T]; L^2(U))$. ## Energy Estimates Before proving existence, we derive the fundamental a priori estimate that any weak solution must satisfy. This is the parabolic analogue of Gårding's inequality in the elliptic theory. [quotetheorem:603] The proof proceeds by testing the weak formulation with $v = u(t)$ itself, using the identity $u_t \circ u = \frac{1}{2}\frac{d}{dt}\|u\|_{L^2}^2$, applying Gårding's inequality for the elliptic coercivity, absorbing the right-hand side via Young's inequality, and closing with Gronwall's inequality. The key mechanism — testing with the solution itself to extract an energy identity — is the same as in the elliptic theory, but the time derivative produces a differential inequality rather than an algebraic one. [remark: Uniqueness] The energy estimate immediately gives uniqueness: if $u_1, u_2$ are both weak solutions with the same data $(f, g)$, then $w := u_1 - u_2$ solves $w_t + Lw = 0$ with $w(\cdot, 0) = 0$. The energy estimate with $f = 0$ and $g = 0$ gives $\|w\|_{\mathcal{W}} = 0$, so $u_1 = u_2$. [/remark] ## Existence of Weak Solutions: Galerkin's Method In the elliptic theory, existence was established via the Lax-Milgram theorem — a direct application of [Hilbert space](/page/Hilbert%20Space) geometry. For parabolic equations, the situation is more involved because the solution lives in a space of time-dependent functions and we must solve an evolution problem, not a fixed equation. The approach is **Galerkin's method**: approximate the infinite-dimensional PDE by a [sequence](/page/Sequence) of finite-dimensional ODE systems, solve each exactly, and pass to the [limit](/page/Limit) using compactness. {width=85%} [motivation] ### The Strategy 1. **Choose a basis.** Let $\{w_k\}_{k=1}^\infty$ be an orthonormal basis of $L^2(U)$ consisting of eigenfunctions of the symmetric elliptic operator (or, more generally, any orthogonal basis of $H^1_0(U)$). These are smooth by the [elliptic regularity theory](/page/Second-Order%20Elliptic%20Equations). 2. **Finite-dimensional approximation.** For each $m \in \mathbb{N}$, seek an approximate solution of the form: \begin{align*} u_m(x, t) := \sum_{k=1}^m d_m^k(t)\,w_k(x), \end{align*} where the coefficients $d_m^k: [0, T] \to \mathbb{R}$ are determined by requiring the PDE to hold when tested against each basis function $w_1, \ldots, w_m$. 3. **ODE system.** The weak formulation $(u_m)_t \circ w_k + B[u_m, w_k; t] = f(t) \circ w_k$ becomes a system of $m$ linear ODEs for $d_m^1(t), \ldots, d_m^m(t)$. Standard ODE theory (Picard-Lindelöf) guarantees existence and uniqueness on $[0, T]$. 4. **Energy estimates.** The a priori energy estimate from the previous section applies to each $u_m$ (the proof only uses the tested weak formulation, which holds by construction). The bound is **uniform in $m$**. 5. **Passage to limit.** The uniform bounds imply that $\{u_m\}$ is bounded in $L^2(0,T; H^1_0(U))$ and $\{(u_m)_t\}$ is bounded in $L^2(0,T; H^{-1}(U))$. By the Banach-Alaoglu theorem, a subsequence converges weakly. The **Aubin-Lions compactness lemma** (which upgrades [weak convergence](/page/Weak%20Convergence) in $L^2(0,T; H^1_0)$ to strong convergence in $L^2(0,T; L^2)$) allows passage to the limit in the bilinear form. [/motivation] [quotetheorem:616] ## Regularity As in the elliptic case, the weak solution $u \in \mathcal{W}$ is initially known only to have one spatial derivative in $L^2$. If the data is smoother, we expect the solution to be smoother. The regularity theory proceeds in exactly the same spirit as for [elliptic equations](/page/Second-Order%20Elliptic%20Equations) — using difference quotients in the spatial variables — but now with additional care for the time variable. [quotetheorem:611] The proof follows the same difference-quotient strategy as the elliptic case: test the weak formulation with $v = -D_k^{-h}(\zeta^2 D_k^h u)$ where $\zeta$ is a spatial cutoff, and derive uniform bounds on the $L^2$ norm of $D_k^h \nabla u$, which imply $u \in H^2$ locally. The parameter $\delta > 0$ is necessary because the initial data $g$ may lack the regularity needed at $t = 0$. [quotetheorem:612] A striking feature of parabolic equations, absent in the elliptic theory, is the **instantaneous smoothing effect**: even if the initial data $g$ is merely $L^2$, the solution $u(\cdot, t)$ becomes $C^\infty$ for every $t > 0$ (provided the coefficients and domain are smooth). This is because the exponential decay of the high-frequency modes (eigenvalues $\lambda_k \to \infty$ in the spectral decomposition) instantly suppresses all irregularity. {width=80%} ## Maximum Principles As in the [elliptic case](/page/Second-Order%20Elliptic%20Equations), maximum principles provide pointwise control on solutions. The parabolic versions are more subtle because the domain of dependence is one-sided in time: the value $u(x_0, t_0)$ depends on the solution at earlier times but not later times. Consequently, the maximum over $\overline{U_T}$ must be attained on the parabolic boundary $\Gamma_T$ — the "bottom" and "sides" of the cylinder, but not the "top." We work with the operator in **nondivergence form**, exactly as in the elliptic maximum principle setting: \begin{align*} Lu = -\sum_{i,j=1}^n a_{ij}(x,t)\,\partial_{x_ix_j} u + \sum_{i=1}^n b_i(x,t)\,\partial_{x_i} u + c(x,t)\,u, \end{align*} and assume $u \in C^2_1(U_T) \cap C(\overline{U_T})$, meaning $u$ is twice continuously [differentiable](/page/Derivative) in $x$ and once in $t$ inside $U_T$, and continuous up to the boundary. ### The Weak Maximum Principle [quotetheorem:607] The proof is a perturbation argument. One first handles the strict case $u_t + Lu < 0$: at any interior maximum, the spatial Hessian is non-positive and the time derivative is non-negative, so the ellipticity and sign conditions force $u_t + Lu \ge 0$ — a contradiction. The general case $u_t + Lu \le 0$ follows by perturbing to the strict subsolution $u^\varepsilon(x,t) = u(x,t) - \varepsilon t$ and sending $\varepsilon \to 0$. ### The Strong Maximum Principle [quotetheorem:613] The strong maximum principle asserts that a non-constant subsolution cannot attain its maximum at any interior space-time point. Moreover, the conclusion is stronger than in the elliptic case: $u$ must be constant on the **entire backward time slab** $\overline{U} \times [0, t_0]$, not just in a spatial neighbourhood. This reflects the parabolic propagation of information: if $u$ achieves its maximum at time $t_0$, it must have been at that value for all earlier times. [remark: Comparison With Elliptic Maximum Principles] The parabolic maximum principles reduce to the elliptic ones for time-independent solutions. If $u(x,t) = u(x)$ does not depend on $t$, then $u_t = 0$ and the condition $u_t + Lu \le 0$ becomes $Lu \le 0$, recovering the [elliptic weak and strong maximum principles](/page/Second-Order%20Elliptic%20Equations). The key new feature is the asymmetry in time: the maximum propagates backward (to earlier times) but not forward. Physically, if the temperature achieves its maximum at time $t_0$, diffusion would have decreased it before $t_0$ unless it was already at that value — hence constancy on $[0, t_0]$. But the temperature at times $t > t_0$ is unconstrained by the maximum at $t_0$. [/remark] ## Harnack's Inequality The Harnack inequality for parabolic equations has a striking feature not present in the elliptic version: it relates the solution at an **earlier** time to the solution at a **later** time, reflecting the causal structure of diffusion. {width=75%} [quotetheorem:614] The crucial point is the **time ordering**: the supremum is at the earlier time $t_1$ and the infimum at the later time $t_2 > t_1$. The inequality says that heat diffuses: if $u$ is large somewhere at time $t_1$, it must still be bounded below everywhere (in $V$) at the later time $t_2$. The reverse inequality (bounding $u(\cdot, t_2)$ by $u(\cdot, t_1)$) is false in general — heat can concentrate. ## Examples [example: The Heat Equation on a Bounded Domain] The simplest parabolic equation is the heat equation with homogeneous Dirichlet conditions: \begin{align*} \begin{cases} u_t - \Delta u = 0 & \text{in } U \times (0, T], \\ u = 0 & \text{on } \partial U \times [0, T], \\ u(\cdot, 0) = g & \text{in } U. \end{cases} \end{align*} Here $L = -\Delta$ with $a_{ij} = \delta_{ij}$, $b_i = 0$, $c = 0$, and $f = 0$. **Eigenfunction expansion.** Let $\{\lambda_k, w_k\}_{k=1}^\infty$ be the eigenvalues and $L^2$-orthonormal eigenfunctions of $-\Delta$ on $U$ with Dirichlet conditions (as constructed in the [spectral theory of elliptic operators](/page/Second-Order%20Elliptic%20Equations)). The Galerkin method with this basis produces exact solutions: the $m$-th approximation $u_m(x,t) = \sum_{k=1}^m d_m^k(t) w_k(x)$ satisfies the ODE $\dot{d}_m^k + \lambda_k d_m^k = 0$, giving $d_m^k(t) = \hat{g}_k e^{-\lambda_k t}$, where $\hat{g}_k = (g, w_k)_{L^2}$. The full solution is: \begin{align*} u(x, t) = \sum_{k=1}^\infty \hat{g}_k\,e^{-\lambda_k t}\,w_k(x). \end{align*} Since $\lambda_k \to \infty$, the exponential decay $e^{-\lambda_k t}$ ensures this [series](/page/Series) converges in every Sobolev norm for $t > 0$, confirming the smoothing effect: even rough initial data $g \in L^2(U)$ produces a $C^\infty$ solution for $t > 0$. The solution converges to zero exponentially as $t \to \infty$, at a rate governed by the first eigenvalue $\lambda_1$. [/example] [example: Separation of Variables and Parabolic Regularity] Consider the heat equation on $U = (0, \pi) \subset \mathbb{R}$ with initial data $g(x) = x(\pi - x)$: \begin{align*} \begin{cases} u_t - u_{xx} = 0 & \text{in } (0, \pi) \times (0, \infty), \\ u(0,t) = u(\pi,t) = 0, \\ u(x, 0) = x(\pi - x). \end{cases} \end{align*} The eigenfunctions of $-\partial_{xx}$ on $(0, \pi)$ with Dirichlet conditions are $w_k(x) = \sqrt{2/\pi}\sin(kx)$ with eigenvalues $\lambda_k = k^2$. Computing the Fourier coefficients: \begin{align*} \hat{g}_k = \sqrt{\frac{2}{\pi}}\int_0^\pi x(\pi - x)\sin(kx)\,dx = \begin{cases} \frac{4\sqrt{2/\pi}}{k^3} & k \text{ odd}, \\ 0 & k \text{ even}. \end{cases} \end{align*} The solution is: \begin{align*} u(x,t) = \frac{8}{\pi}\sum_{\substack{k=1 \\ k \text{ odd}}}^\infty \frac{1}{k^3}\,e^{-k^2 t}\,\sin(kx). \end{align*} The initial data $g \in H^1_0(0, \pi)$ but $g \notin H^2(0, \pi) \cap H^1_0(0, \pi)$ (since $g''(x) = -2$ does not vanish at the boundary, violating compatibility). For $t > 0$, however, the exponential decay of $e^{-k^2 t}$ ensures the series converges in every $H^m$, confirming the instantaneous smoothing predicted by the regularity theorem. [/example] ## Problems [problem] **(Energy identity.)** Let $u \in \mathcal{W}$ be the weak solution of $u_t - \Delta u = 0$ in $U_T$ with $u(\cdot, 0) = g \in L^2(U)$ and $u = 0$ on $\partial U$. 1. Prove the energy identity: \begin{align*} \|u(t)\|_{L^2(U)}^2 + 2\int_0^t \|\nabla u(s)\|_{L^2(U)}^2\,ds = \|g\|_{L^2(U)}^2 \quad \text{for all } t \in [0, T]. \end{align*} 2. Deduce that $\|u(t)\|_{L^2(U)}$ is non-increasing in $t$. 3. Use the Poincaré inequality $\|v\|_{L^2} \le C_P \|\nabla v\|_{L^2}$ for $v \in H^1_0(U)$ to show exponential decay: $\|u(t)\|_{L^2(U)} \le e^{-t/C_P^2}\|g\|_{L^2(U)}$. [/problem] [solution] **Part 1.** Set $v = u(t)$ in the weak formulation $u_t \circ v + (\nabla u, \nabla v)_{L^2} = 0$: \begin{align*} u_t(t) \circ u(t) + \|\nabla u(t)\|_{L^2}^2 = 0. \end{align*} The identity $u_t \circ u = \frac{1}{2}\frac{d}{dt}\|u\|_{L^2}^2$ (justified by the Gelfand triple structure) gives: \begin{align*} \frac{1}{2}\frac{d}{dt}\|u(t)\|_{L^2}^2 + \|\nabla u(t)\|_{L^2}^2 = 0. \end{align*} Integrating from $0$ to $t$: \begin{align*} \frac{1}{2}\|u(t)\|_{L^2}^2 - \frac{1}{2}\|g\|_{L^2}^2 + \int_0^t \|\nabla u(s)\|_{L^2}^2\,ds = 0, \end{align*} which rearranges to the energy identity. **Part 2.** Since $\int_0^t \|\nabla u\|_{L^2}^2\,ds \ge 0$, the energy identity gives $\|u(t)\|_{L^2}^2 \le \|g\|_{L^2}^2$ for all $t$. More generally, replacing $0$ by $s < t$: $\|u(t)\|_{L^2}^2 \le \|u(s)\|_{L^2}^2$, so $t \mapsto \|u(t)\|_{L^2}$ is non-increasing. **Part 3.** The Poincaré inequality gives $\|\nabla u\|_{L^2}^2 \ge C_P^{-2}\|u\|_{L^2}^2$. Substituting into the differential identity: \begin{align*} \frac{d}{dt}\|u\|_{L^2}^2 = -2\|\nabla u\|_{L^2}^2 \le -\frac{2}{C_P^2}\|u\|_{L^2}^2. \end{align*} This is the ODE $\varphi'(t) \le -\frac{2}{C_P^2}\varphi(t)$ for $\varphi(t) = \|u(t)\|_{L^2}^2$. Gronwall's inequality gives $\varphi(t) \le e^{-2t/C_P^2}\varphi(0)$, i.e., $\|u(t)\|_{L^2} \le e^{-t/C_P^2}\|g\|_{L^2}$. [/solution] [problem] **(Parabolic maximum principle.)** Let $u \in C^2_1(U_T) \cap C(\overline{U_T})$ satisfy $u_t - \Delta u = 0$ in $U_T := (0,1) \times (0,1]$. 1. Suppose $u(0, t) = u(1, t) = 0$ for all $t \in [0, 1]$, and $u(x, 0) = \sin(\pi x)$. Without solving the equation explicitly, use the maximum principle to prove that $0 \le u(x, t) \le 1$ for all $(x, t) \in \overline{U_T}$. 2. Use the strong maximum principle to show that the inequality is strict: $0 < u(x, t) < 1$ for all $(x, t) \in U_T$. [/problem] [solution] **Part 1.** The function $u$ satisfies $u_t + Lu = 0$ with $L = -\partial_{xx}$, $c = 0 \ge 0$. The parabolic boundary data satisfies $0 \le u \le 1$ on $\Gamma_T$: on the lateral boundary, $u = 0$; on the initial slice, $u(x, 0) = \sin(\pi x) \in [0, 1]$ for $x \in [0, 1]$. For the upper bound: $u_t + Lu = 0 \le 0$, so $u$ is a subsolution. The weak maximum principle gives $\max_{\overline{U_T}} u = \max_{\Gamma_T} u = 1$. For the lower bound: $-u$ satisfies $(-u)_t + L(-u) = 0 \le 0$, so $-u$ is also a subsolution. The weak maximum principle gives $\max_{\overline{U_T}}(-u) = \max_{\Gamma_T}(-u) = 0$, i.e., $\min_{\overline{U_T}} u = 0$. **Part 2.** The initial data $g(x) = \sin(\pi x)$ is strictly positive on $(0, 1)$ and vanishes only at $x = 0$ and $x = 1$. By the strong maximum principle, if $u$ attained its maximum value $1$ at an interior point $(x_0, t_0) \in U_T$, then $u$ would be constantly $1$ on $\overline{U} \times [0, t_0]$. But $u(0, t) = 0 \neq 1$, a contradiction. So $u < 1$ strictly in $U_T$. Similarly, if $u$ attained its minimum $0$ at an interior point, $-u$ would attain its maximum there, forcing $-u$ (hence $u$) to be constant $0$ on the backward time slab, contradicting $u(x, 0) = \sin(\pi x) > 0$ for $x \in (0, 1)$. So $u > 0$ strictly in $U_T$. [/solution] ## References 1. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 7. 2. L. C. Evans, *Partial Differential Equations*, 2nd ed., AMS (2010). Ch. 6 (for the elliptic theory used throughout). 3. J. Wloka, *Partial Differential Equations*, Cambridge University Press (1987). Ch. IV. 4. A. Friedman, *Partial Differential Equations of Parabolic Type*, Dover (2008). 5. O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'ceva, *Linear and Quasi-linear Equations of Parabolic Type*, AMS Translations (1968).