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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.

See also:

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.
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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.
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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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