The author considers the uniqueness of subsolution to the heat equation on Finsler manifold.
Let \((M,F)\) be a Finsler manifold with a Finsler metric \(F: TM \to [0,+\infty)\) associated with the Minkowski norm on each fiber \(T_xM, x\in M\), homogeneous on the fiber variable \(F(x,\lambda y) = \lambda F(x,y), \forall \lambda > 0\), and such that the matrix \((g_{ij}(x,y)) := (\frac{1}{2}(F^2)_{y^iy^j})\) is positive definite. A Finsler measure space \((M,F,m)\) is a Finsler space \((M,F)\) with a measure \(m\). The nonreversibility \(F(x,y) \ne F(x,-y)\) causes an asymmetry of the associated distance. One defines the dual metric \[F^*(x, \xi) = \sup_{F(x,y) = 1} \xi(y),\] and as usual \(L^p\) spaces and Sobolev spaces \[L^p = \{ u ; \int_M [F^*(du)]^p dm < \infty \}, W^{1,p} = \{ u\in L^p(M)\; |\; ||u||_{L^p} + ||F^*(du)||_{L^p} + ||F^*(-du)||_{L^p} < \infty \},\] \[W^{1,p}_0 = \overline{[C^\infty_0(M)]_{1,p}}\] as the completion of the space of smooth functions with compact support with respect to the Sobolev norm \(||.||_{1,p}\), \(H^1 := W^{1,2}\), the Banach dual \(H^{-1} := (H^1)^*\). A subsolution \(u\) on \([0,T] \times M\) to the heat equation \(\partial_t u = \Delta u\) is a function \[u \in L^2([0,T],H^1(M)) \cap H^1([0,T], H^{-1}(M))\] with \(\int_M \Phi \partial_t u dm < - \int_M d\Phi (\nabla u)dm\).
The main results are:
1. For \(p>1\), any subsolution \(u(t,x)\) to the heat equation on \([0,\infty) \times M\) satisfying \(u(t,x) \in L^p(M)\) and \[\int_M u^p(t,x) dm \equiv 0, \forall t>0\] must vanish identically (Theorem 1.1).
2. If the Ricci curvature is nonnegative, Ric\(_N. \geq 0\) for some \(N \in [n,\infty)\), then for any \(0<p\leq 2\) any nonnegative subsolution \(u \in L^p(\mathbb R \times M)\) on \(\mathbb R \times M\) must vanish identically (Theorem 1.2).