Bifurcation for a free boundary problem modeling tumor growth with ECM and MDE interactions

We study a free boundary problem modeling solid tumor growth. The simplified model contains a parameter lambda. Different from previous works on bifurcation analysis, a new ingredient of the present paper is that the influence of the extracellular matrix (ECM) and matrix degrading enzymes (MDE) interactions is included in the model. We first show that for each lambda > 0, there exists a unique radially symmetric stationary solution with radius tau = R-S. Then we prove that there exist a positive integer n* and a sequence of lambda(n) (n > n*) for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric one. In particular, we discover that these lambda(n) are larger than those (lambda) over tilde (n) previously known when the effects of ECM and MDE are not considered in the model. (C) 2018 Elsevier Ltd. All rights reserved.