I've played with such things 15 years ago. Here's what I remember...

Sure you need some semi-boundedness condition on K. Uniqueness should then follow via semigroup domination (aka Kato's inequality) from scalar case.

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108

Shigekawa had rediscovered Barry Simon's semigroup domination criterion (i.e. Kato's inequality) and generalized it to the vector valued things.

R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, J. Funct. Analysis 52 (1983), 48-79. Scalar case: Theorem 3.5 - but has uniqueness only for 1< p<∞.

(Strichartz' Some details in Strichartz' proof(s) for vector valued Laplacians not optimal , as far as I remember, (classic problems with doing tensor calculus and Stokes...) not even in the follow-up paper "L^p contractive projections and the heat semigroup for differential forms" J. Funct. Analysis, 65 (1985), 348-357)

Perhaps look also for El Maati Ouhabaz ca. 1999. He's written a book, "Analysis of Heat Equations on Domains" (2004) but I never got hold of it.

I've played with such things 15 years ago. Here's what I remember...

Sure you need some semi-boundedness condition on K. Uniqueness should then follow via semigroup domination (aka Kato's inequality) from scalar case.

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108

R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, J. Funct. Analysis 52 (1983), 48-79. Scalar case: Theorem 3.5 - but has uniqueness only for 1< p<∞.

Strichartz'

(Strichartz' proof(s) for vector valued Laplacians not optimal, as far as I remember, not even in the follow-up paper "L^p contractive projections and the heat semigroup for differential forms" J. Funct. Analysis, 65 (1985), 348-357

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108)

Perhaps look also for El Maati Ouhabaz ca. 1999. He's written a book, "Analysis of Heat Equations on Domains" (2004) but I never got hold of it.

2 added 28 characters in body

R.S. Strichartz, Analysis of the Laplacian on the Complete Riemannian Manifold, J. Funct. Analysis 52 (1983), 48-79. Scalar case: Theorem 3.5 - but has uniqueness only for 1< p<∞.

(Strichartz'

Strichartz' proof(s) for vector valued Laplacians not optimal, as far as I remember, not even in the follow-up paper "L^p contractive projections and the heat semigroup for dierential differential forms" J. Funct. Analysis, 65 (1985), 348-357)

I. Shigekawa, L^p Contraction Semigroups for Vector Valued Functions, J. Funct. Analysis 147 (1997), 69-108

Perhaps look also for El Maati Ouhabaz ca. 1999. He's written a book, "Analysis of Heat Equations on Domains" (2004) but I never got hold of it. Also I. Shigekawa ca. 1997 on vector valued semigroups might have useful hints even in scalar case.

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